Coherent structures characterization in turbulent flow

COHERENT STRUCTURES
CHARACTERIZATION IN TURBULENT FLOW
ALEX LIBERZON

COHERENT STRUCTURES CHARACTERIZATION IN
TURBULENT FLOW
RESEARCH THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF SCIENCE
ALEX LIBERZON
SUBMITTED TO THE SENATE OF THE TECHNION — ISRAEL INSTITUTE OF TECHNOLOGY
KISLEV, 5763 HAIFA NOVEMBER, 2002

Contents
1 Introduction 1
1.1 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Literature review 3
2.1 Investigation of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Coherent structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Coherent structures in boundary layers . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Conceptual models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 PIV-based techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Hot-Foil Infrared Imaging Technique (HFIR) . . . . . . . . . . . . . . . . . . 10
2.4 Coherent structure identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.2 Discriminant-based structure identification . . . . . . . . . . . . . . . . . . . 12
2.4.3 Vortex-induced pressure minimum identification . . . . . . . . . . . . . . . . 13
2.4.4 Statistical expansion of the velocity field: the proper orthogonal decomposition 14
2.4.5 Practical implementation of POD . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.6 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Conditional sampling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.1 u − v quadrant technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.2 Variable-interval-time-average technique . . . . . . . . . . . . . . . . . . . . . 20
2.5.3 Variable Interval Space Averaging (VISA) . . . . . . . . . . . . . . . . . . . . 20
ii

CONTENTS iii
2.5.4 Vorticity based identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.5 Detection: Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.6 Eduction: Ensemble average . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.7 Realignment: Signal Enhancement . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.8 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.9 Pattern recognition techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.10 Comparison between different conditional sampling techniques . . . . . . . . 24
2.5.11 Recent work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Mathematical background 26
3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Mean flow equations and Reynolds stress . . . . . . . . . . . . . . . . . . . . 28
3.1.3 Reynolds stress and vortex stretching . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Kinetic energy of the mean and turbulent flow . . . . . . . . . . . . . . . . . . . . . 29
3.3 Vorticity and velocity gradient tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Analysis approach 31
4.1 Decomposition of turbulent flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Guidelines of the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Experimental apparatus 34
5.1 The infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 Stereoscopic PIV system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2.1 Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2.2 Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2.3 Acquisition and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2.4 Synchronization and processing . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4 XPIV – Multi-plane Stereoscopic Particle Image Velocimetry . . . . . . . . . . . . . 41
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

CONTENTS iv
5.5 Optical arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5.1 Variable light intensity scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5.2 Variable inter-plane distance scheme . . . . . . . . . . . . . . . . . . . . . . . 44
5.5.3 Calibration of XPIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.6 Image processing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.6.1 Pre-processing of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.6.2 Particle images in the plane of focus . . . . . . . . . . . . . . . . . . . . . . . 47
5.6.3 Discrimination between two defocus planes . . . . . . . . . . . . . . . . . . . 49
5.7 Combined PIV and HFIR experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.7.1 Preprocessing of the thermal images . . . . . . . . . . . . . . . . . . . . . . . 55
6 Results and discussion 59
6.1 Conventional turbulent boundary layer flow analysis, x1 − x2 plane . . . . . . . . . 59
6.1.1 Velocity fields and distributions . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Linear combination of the POD modes . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.1 POD of velocity/vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2.2 Comparison of the POD of the velocity and vorticity . . . . . . . . . . . . . 68
6.2.3 Average field and the first POD mode . . . . . . . . . . . . . . . . . . . . . . 68
6.2.4 Symmetry of the orthogonal decomposition . . . . . . . . . . . . . . . . . . . 72
6.2.5 Choice of eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.6 Vorticity component ω3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2.7 Parametrization of the results . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.8 Identified coherent structure - discussion . . . . . . . . . . . . . . . . . . . . 78
6.2.9 Results from the x1 − x3 plane . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.10 Results from the x2 − x3 plane . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2.11 Characterization of the structure by using the three-dimensional reconstruc-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 XPIV results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Coherent structures in XPIV results . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5 Characterization of DNS data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

CONTENTS v
6.6 Combined velocity/temperature footprints . . . . . . . . . . . . . . . . . . . . . . . 83
7 Summary and Conclusions 91
7.1 Spatial characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A Error estimation of PIV experimental data 94
A.1 Mean velocity confidence level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.2 Confidence limits of standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . 95
A.3 Error estimation at the measured velocity from the PIV . . . . . . . . . . . . . . . . 96
B PIV validation 98
B.1 Flow rate comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
B.2 Software performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B.3 Comparative measurements with PIV and LDV systems . . . . . . . . . . . . . . . . 101
B.4 Out-of-plane component validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.5 Velocity derivatives, calculation and validation . . . . . . . . . . . . . . . . . . . . . 103
C Derivatives. Part 1: Vorticity calculation 105
C.0.1 Standard differentiation schemes . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.0.2 Alternative differentiation schemes . . . . . . . . . . . . . . . . . . . . . . . . 106
C.0.3 Uncertainties and errors in differential estimation . . . . . . . . . . . . . . . . 108
C.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.1.1 The test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.1.2 Numerical error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.2 Appendix B - Impinging Jet Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C.3 Appendix B - Matlab r
procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A XPIV - Image processing definitions 124
A.1 Percentile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

CONTENTS vi
A.1.1 Histogram based operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.1.2 Derivative based operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2 Morphology based operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.3 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B Surfactants 129
C Principles of PIV measurement technique 137
D Principles of Stereo PIV 139
D.1 Particle imaging geometry reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 139
D.2 Different SPIV configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

List of Figures
5.1 Facility schematic view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Illumination scheme in two views: front (left) and top (right) . . . . . . . . . . . . . 36
5.3 Stereoscopic PIV configuration scheme for the x1 − x2 plane experiment. . . . . . . . 39
5.6 Schematic view of the first type of the optical array. . . . . . . . . . . . . . . . . . . 42
5.7 Schematic view of the second type of the optical array. . . . . . . . . . . . . . . . . . 44
5.8 Original three plane PIV image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.9 Enhanced three plane PIV image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.10 Schematic view of the reconstruction principle used in the region growing algorithm:
(- -) Dashed line shows the one dimensional signal, (− · −) line is for the identified
saturated pixels and, (− • −) line presents the reconstructed object. . . . . . . . . . 48
5.11 Image with particles in the focus plane. . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.12 Original (left) and defocus planes image (right). . . . . . . . . . . . . . . . . . . . . . 50
5.13 Defocus planes image (left) and gradient map as a gray level image (right). . . . . . 51
5.14 Gradient image (left) and enhanced gradient map (right) . . . . . . . . . . . . . . . . 51
5.15 Defocus particles image (left) and the identified objects in a binary image (right). . . 52
5.16 Size distribution (granulometry) of the binary image (left) and its derivative (right). 53
5.17 Schematic view of the combined PIV - HFIR experimental setup. . . . . . . . . . . . 54
5.18 Infrared image of the temperature field of the foil surface . . . . . . . . . . . . . . . 56
5.19 Trimmed image of the temperature field. . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.20 Enhanced image of the temperature field. . . . . . . . . . . . . . . . . . . . . . . . . 57
vii

LIST OF FIGURES viii
5.21 Temperature field image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.1 Example of the instantaneous fluctuating velocity field, u1,2. . . . . . . . . . . . . . . 60
6.2 Instantaneous profiles of streamwise ũ1 (left) and spanwise ũ2 velocity components
(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 Ensemble averaged velocity field. Note the streamwise velocity profile. . . . . . . . . 62
6.4 Streamwise velocity distributions in wall units, along with the log-law line. . . . . . . 63
6.5 Joint PDF between u1 and u2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.6 Reynolds stress hu1u2i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.7 Streamwise kinetic energy u2
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.8 Wall normal kinetic energy u2
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.9 Turbulent kinetic energy production −2hu1u2iS12 . . . . . . . . . . . . . . . . . . . . 65
6.10 Turbulent kinetic energy production −2hu1u2iS12 versus wall normal coordinate. . . 66
6.11 Ensemble averaged vorticity ω3 (left) and strain S12 (right). . . . . . . . . . . . . . . 66
6.12 Instantaneous vorticity ω3 component field. . . . . . . . . . . . . . . . . . . . . . . . 67
6.13 First POD mode of the fluctuating velocity field (left) and vorticity (right) . . . . . 69
6.14 Second POD mode of the fluctuating velocity field (top) and vorticity (bottom) . . . 70
6.15 POD modes of the instantaneous (ω̃3, left) and fluctuating (ω3, right) vorticity fields. 71
6.16 Symmetric modes evidence in ”energy” spectrum of the decomposition. . . . . . . . 73
6.17 Four (a), fifth (b), six (c), and linear combination of 5th and 6th (d) POD modes of
the fluctuating vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.18 Linear combination of the POD modes of the fluctuating vorticity ωz component, a)
3, b) 5, c) 10, and d) 150 modes, respectively. . . . . . . . . . . . . . . . . . . . . . . 75
6.19 Linear combination of three POD modes of the vorticity component, ω3. . . . . . . . 76
6.20 Linear combination of three POD modes of the rate of strain component S12. . . . . 77
6.21 Linear combination of three orthogonal modes of the vorticity ω3 for a) Reh = 24000,
b) Reh4 = 27000. Linear combination of three orthogonal modes of the vorticity ω3
for c) Reh = 45000, d) Reh4 = 54000. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.22 Contour map of the streamwise velocity fluctuations u1 along with the vector plot of
the velocity fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

LIST OF FIGURES ix
6.23 Instantaneous field of the streamwise velocity fluctuations (red and blue line contours)
over the field of the ∂(u1u3)
∂x3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.24 Linear combination of three POD modes of the wall normal vorticity component ω2
for x2/h = 0.3125 (top plane), 0.125 (middle), and 0.0375 (bottom plane). . . . . . . 87
6.25 Linear combination of the POD modes of the streamwise vorticity ω1 component in
the x2 − x3 plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.26 Schematic view of the POD modes combinations as the projections on three orthog-
onal planes. Note that x − z plane is at y+
= 100, the y+
axis is for the x − y and
y − z planes only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.27 Streamwise velocity average profiles measured by using XPIV (-o) and box-plot of the
PIV measurements in separate y planes(|-[]-|). . . . . . . . . . . . . . . . . . . . . 89
6.28 Relative turbulent intensities u1/U1 and u3/U1 for planes from the XPIV and 2D PIV
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.29 Distribution of η̄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.1 Standard PIV image and the analysis with InsightTM
software . . . . . . . . . . . . 100
B.2 Comparison of the results calculated by InsightTM
and URAPIV software. . . . . . 100
B.3 Schematic view of the PIV and LDV measurement systems and flow configuration. . 102
B.4 Velocity results of the PIV versus LDV measurement results. . . . . . . . . . . . . . 103
B.5 Velocity profile measured in x − y (+) and in y − z (•) configurations. . . . . . . . . 104
C.1 Taylor-Green vortex flow field and its vorticity. . . . . . . . . . . . . . . . . . . . . . 111
C.2 Relative error as a function of simulation runs number for (a) 2%, (b) 5% and (c)
7.5% velocity noise level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
C.3 Relative error as a function of additive noise level for (a) 100, (b) 500, and (c) 1000
simulation runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.4 Mean value of the out-of-plane strain estimation (the mean error of the continuity
equation) for the 500 simulation runs and 5% additive noise level. The plot consists
of the results for the ’Center’ - upper left, ’Richardson’ - upper right, ’Least Squares’
- lower left, and ’Circulation’ calculation scheme at the lower right corner. . . . . . . 114

LIST OF FIGURES x
C.5 Difference between the average of vorticity fields and the vorticity of the average
velocity field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C.6 Average vorticity field of the impinging jet flow. . . . . . . . . . . . . . . . . . . . . . 116
B.1 schematic drawing of the experimental setup (top) and piv configuration (bottom) . 130
B.2 ensemble average of the turbulent intensity
p
hu2
1i/uq for the water (top) and surfac-
tant solution (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.3 ensemble average of the one-point correlation between streamwise and spanwise ve-
locity fluctuations hu1u3i for the water (top) and surfactant solution (bottom). . . . 134
B.4 streamwise average of the hu1u3i correlation for the water (solid line) and surfactant
solution (star-marked line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.5 ensemble average of the turbulent kinetic energy production term hu1u3is13 for the
water (top) and surfactant solution (bottom). . . . . . . . . . . . . . . . . . . . . . . 136
C.1 The principle scheme of the PIV measurement technique. . . . . . . . . . . . . . . . 137
C.2 The measurement system devices and their general arrangement. . . . . . . . . . . . 138
D.1 Scheme of the particle displacement imaging process . . . . . . . . . . . . . . . . . . 140
D.2 Schematic view of the translation SPIV system configuration with shifted imaging
and optical axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
D.3 Schematic view of the stereoscopic PIV imaging system . . . . . . . . . . . . . . . . 142

List of Tables
5.1 Experimental parameters of the StereoPIV. . . . . . . . . . . . . . . . . . . . . . . . 41
B.1 Comparison of the flow rate estimated by PIV measurements and directly measured
by flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.1 First order differential operators for data spaced at uniform intervals . . . . . . . . . 107
xi

Chapter 1
Introduction
”Turbulence is the most important unsolved problem of classical physics.”
(Feynman, 1965)
Turbulence and turbulent flows continue to magnetize the investigators as a problem that still
requires understanding and the prediction of behavior. Starting from sketches of the turbulent flows
of Leonardo Da Vinci (1510), people repeatedly put forth effort to describe the flow properties.
The study of coherent structures embedded in turbulent boundary layer flows is important to
understand the dynamics and the transfer processes of momentum, heat and mass in most flows of
engineering interest. Also, the automatic and continuous monitoring and identification of the struc-
tures present in such flows could be used to control turbulence. For instance, the most meaningful
energy saving result could be achieved by the drag forces and pressure gradients reduction, that is
feasible only by the control of the turbulence in the flow.
Across the years, significant headway has been achieved in the learning and describing the tur-
bulent wall-bounded boundary layer flows. Yet, this prototypic flow (turbulent boundary layer) is
far from being adequately understood. Additional comprehension should be achieved by means of
innovative experiments, extensive direct numerical simulations (DNS) and theory developments.
The main goal of this particular research are amplification of the image-processing-based exper-
imental methods and identification post-processing analysis to the level that will give way to the
truly quantitative picture of the turbulent structures near the wall.
1

CHAPTER 1. INTRODUCTION 2
Up to these days, present measurement systems do not allow fully three dimensional description
of the flows, or have not attain the needed temporal resolution for the statistical description of the
turbulent flow. Hence, one of the objectives of this study have been chosen to improve the abilities
of the particle image velocimetry (PIV) measurement technique by itself and by combination with
hot-foil infrared imaging (HFIRI) technique, in order to achieve the experimental characterization
of the three-dimensional features in the turbulent boundary layer flow. The obtained measurements
designated to the extensive post-processing procedure, including the statistical analysis and coherent
structures identification processes. These two main parts of the research would provide the essential
understanding of the turbulent boundary layer flow.
Heat transfer, combustion, chemical reaction, drag and aerodynamic noise generation are fields in
which better understanding of coherent structures should produce substantial technological benefits.
1.1 Current research
The boundary layer turbulent flow have been investigated in a flume. The velocity of the flow has
been measured by using particle image velocimetry and assisted by the hot-foil infrared imaging tech-
nique. The cooperative usage of these measurement methods provides an extensive characterization
of the flow field at both near- and far-wall regions.
The measured velocity field, its numerically calculated velocity gradient tensor and following
vorticity vector fields have been served the characterization analysis. Such analysis provides the
kinematic description of the flow, based on the symmetric and anti-symmetric parts of the gradient
tensor (i.e. the rate-of-deformation and rate-of-rotation tensors), stresses, and different turbulent
characteristics, for instance: turbulence intensity, Reynolds stresses, length scale and other correla-
tions. Moreover, a full description of the turbulent boundary layer flow have supplied the information
needed to accurately identify coherent and non-coherent parts of the turbulent flow by conditional
(phase) averaging techniques, pattern recognition and decomposition techniques.

Chapter 2
Literature review
The literature review presented here contains three main themes: (i) short survey of turbulence
and turbulent structures research, (ii) a review of experimental techniques used for the turbulence
research, and (iii) survey of structure identification studies.
2.1 Investigation of turbulence
Turbulent flow and turbulent boundary layer flow near walls are common in a wide variety of
applications, including external flows on aircraft and ship surfaces and internal flows in pipes, ducts
and channels. In practically all situations, small disturbances in the flow field, which might be
previously considered irrotational and laminar, are amplified and in the process, which is not entirely
understood, boundary layer go through the transition to the fully turbulent state.
Despite the considerable effort which has been expanded in attempt to develop theory to explain
and experimental methods to measure and understand this complex phenomenon, the general picture
is still unclear. Up to today, there is no measurement technique which allow measuring of flow
characteristics in three-dimensional and temporal (i.e. four-dimensional) domain. The numerical
approach to solve directly or through simulation the Navier-Stokes equations, is still limited by
available computer resources to low-Reynolds-number flows or strongly depended on the turbulence
model and numerical scheme it uses.
3

CHAPTER 2. LITERATURE REVIEW 4
2.2 Coherent structures
At first view, turbulent flows seem to be too complicated, particularly near walls. Early experimental
measurements at a point in a turbulent boundary layer seem to suggest that the velocity fluctuates
randomly about some mean value. In that stage of the turbulence research, it was recognized
that the prediction of the details of time-dependent turbulent flow is not possible. Consequently,
the efforts, both theoretical and experimental, concentrated on understanding the mean or time-
averaged quantities behavior, especially for fully-developed turbulent flow in turbulent boundary
layers.
Starting in 1956 and through the early 1960’s a series of experiments by (Kline et al., 1967,
Runstadler et al., 1963), where the flow was visualized using hydrogen bubbles, definitively demon-
strated that the turbulent boundary layer flow is much more structured than had been previously
thought to be true. Unfortunately, the notion of structure in turbulent flow is very controversial
through the years, and ultimate definition of coherent structure does not exist.
As an alternative of the exact definition of coherent structures, one could use the simplest evidence
of the coherency in the flow, i.e., turbulence characteristics have nonzero correlation values one to
another, due to the fact that they are never independently random, and consequently, the turbulence
intensity (its root-mean-square value) cannot be described by normal probability density function.
Moreover, it is impossible to give an account of the spatially connected group of turbulent flow
elements, entitled as ’vortex’ or ’eddy’ by common statistical tools, due to its progressive life cycle
nature (i.e. birth, development, interaction and breakdown life cycle stages). In addition, vortices
have different sizes and shapes and appear randomly in space and time in different turbulent flows.
2.2.1 Coherent structures in boundary layers
Extensive research on turbulent boundary layers performed over the last four decades (see for in-
stance Robinson (1991a)) proposes that the near-wall region is dominated by coherent structures
playing a critical role in the turbulence production, transport and dissipation processes. In early
50’s, Klebanoff (1954) and Laufer (1954) showed that the significance of the near-wall region, where
about 80% of the total dissipation and production takes place.
In very systematic visualization study, using hydrogen bubbles, Kline et al. (1967), Runstadler

et al. (1963) observed the intermittent streaky structure and the occasional ’lifts’ of these low
streamwise momentum streaks and their interaction with the outer flow field. In a subsequent
study, Kim et al. (1971) showed that essentially all the turbulence production occurs during those
bursting periods and in the wall region 0 < y+
< 100 (The wall regions corresponds to y+
= yuτ /ν,
where uτ is the friction velocity and ν the viscosity; the friction velocity is defined as µ∂U/∂y = ρu2
τ ,
with the gradient evaluated at the wall). After these pioneering works, various non-random events
have been experimentally identified, such as wall low-speed streaks, internal shear layers, vortical
structures, ejections and sweeps Smith and Walker (see review of 1995). Thus, Blackwelder and
Eckelmann (1979) studied in details the structure of wall streaks by combination of hot-film and
wall attached sensors. Head and Bandyopadhyay (1981) used smoke visualization of the turbulent
boundary layer structures to observe an eddy-like structures laying on the inclined plane. More
recently, Smith and Metzler (1983), Moin and Kim (1985), and Kim et al. (1987), among others,
confirmed some characteristic properties of streaks, like their spanwise separation distance λ+
≈ 100,
and inclination angle of vortices ≈ 10◦
.
The recent research has been done through advanced experimental techniques, like PIV method
by Meinhart and Adrian (1995), hot-foil infrared imaging technique by Hetsroni et al. (1996, 1997)
among others. These researches investigated in more spatial details the velocity and temperature
fields of the wall-bounded flows, respectively, and gained the description of the turbulent structures
through their two-dimensional velocity/temperature footprints.
2.2.2 Conceptual models
Following the definition proposed by Robinson (1991a), a conceptual model is ’an idealized descrip-
tion of the physical processes underlying the observed behavior of turbulent boundary layers’. Only
a few of the many models proposed in the literature will be reviewed in this section, partially taken
from the reviews collection book edited by Panton (1997) and from a comprehensive review of the
coherent motions provided by Robinson (1991a).
Probably the first conceptual idea of the horseshoe-shaped vortex was proposed by Theodorsen
(1952), and is based on the vorticity form of the Navier-Stokes equation. A model proposed later,
by Townsend (1956), was based on the two-point-correlation measurements at the near-wall region.
The conclusion was that the dominant structures near the wall were tilted streamwise vortices,

and it lately was recognized that this structure appears to be the feet of Theodorsen’s horseshoe
vortices. The later experimental Head and Bandyopadhyay (1981) and direct numerical simulation
Moin and Kim (1985) investigations approved the existence of structures similar to the ’horseshoe-
shaped’ vortices. In addition, Rashidi and Banerjee (1990) noted that rather than horseshoe vortices,
structures which seem like half a horseshoe, or a hockey-stick, are more commonly seen. Willmarth
and Tu (1967) proposed a hairpin-shaped vortical structure sloped downstream at about 10◦
from
the wall. Offen and Kline (1975) suggested that the feature structure is a lifted and stretched
horseshoe-shaped vortex loop, and therefore, the near-wall bursting processes are consistent with
the passage of such horseshoe vortex. Smith (1984), Smith and Walker (1995) proposed a symmetric
hairpin-shaped vortex model that explains the streak-bursting process, from oscillation to hairpin
vortices to sweeps, in terms of interactions of multiples of Theodorsen’s rollup structure.
Robinson (1991a) summarized and grouped the various experimentally observed coherent struc-
tures in the boundary layer in eight categories: low-speed streaks, ejections, sweeps, vortical struc-
tures (hairpin vortices with elongated trailing legs), near-wall shear layers, near-wall pressure pock-
ets, δ-scale shear layers or ’backs’, and large scale motions (horseshoe vortices that are as wide as
their height).
Recent researches, includes the research of Zhou et al. (1999) that extracted the picture of
pockets of hairpin vortices from numerical results, following the Smith’s model. The presented
results appeared to be consistent with the experimental observations of Meinhart and Adrian (1995).
From the other hand, new theoretical model proposed by Levinski and Cohen (1995) describes the
mechanism leading to the quick growth of such hairpin vortices in the turbulent boundary layer.
Several researches proposed recently the ideas of macro-structures in the turbulent boundary layer,
similar to the proposal of Kaftori et al. (1994), which suggested that almost all the processes could
be explained by a large-scale funnel-shaped streamwise vortices.
Unfortunately, there has been much less progress in the development of models capable of han-
dling all observed features in the turbulent boundary layer. Part of the reason is that we still lack a
complete physical picture of what Townsend (1956) called the ”main turbulent motion” and which,
as Klebanoff (1954) showed, contains most of the turbulent kinetic energy and is responsible for
the generation of most of the Reynolds stress. This is not so surprising in view of the difficulties
inherent in the overall description of such a complex, time-dependent, three-dimensional phenomena

as turbulent boundary layer flow.
Bernard and Wallace (1997) (in Panton, 1997) cited the work of Robinson (1991b):
One of the Robinson’s important interpretive observations was that the closely spaced
bundles of vortex lines shaped into horseshoe, hairpin or loop shaped filaments are NOT
necessarily coincident with actual vortices. He noted that such vortex filament loops
result from the distortions to the vorticity field caused by the presence of quasi-streamwise
vortices and are spatially associated with them, but that the loops do not describe the
geometry or even the orientation of the pre-existing vortices.
2.3 Experimental techniques
”And this experiment you will make with a square glass vessel, keeping your eye at about
the center of one of these walls; and in the boiling water with slow movement you may
drop a few grains of panic-grass, because by means of the movement of these grains you
can quickly know the movement of the water that carries them with it. And from this
experiment you will be able to proceed to investigate many beautiful movements which
result from one element penetrating into another (i.e., ’air’ (= steam) into water).”
Leonardo da Vinci (1510).
The first experimental technique that have been widely used to describe flows is flow visualization
method. Early implementations were mostly qualitative visualization techniques, (Hesselink, 1988)
and, like the experiments with smoke and dye line patterns, have produced impressive milestones
in the history of flow analysis. After that, the attention had been turned to the development of
anemometers that provide quantitative velocity records of high accuracy and time resolution. How-
ever, its disadvantage is that the records are available only at one selected location within the flow.
Hot-wire and hot-film anemometers, which are based on thermal probing, have become a useful and
common experimental tool in velocimetry. Unfortunately, their finite dimensions, limits the spatial
resolution that could be achieved and, moreover, the tool actually behaves as a flow disturbance.
Next step was to apply the optical probing, that allows non-contact remote measurements. The
measurements are available within a very small sample volume and, the Laser Doppler velocimetry

(LDV) have become the best choice in single-point time-resolved measurements. Although, the tech-
nique is non-interfering, but it still has a drawbacks which relies on the presence of small particles
and this fact naturally sets the limits on the largest velocity gradients acceptable. Moreover, the
general tool is available to measure usually only one velocity component. The extension to three-
component is available, but still is very complex and costly equipment, and only provides data from
a single point in a space.
Recently, the method that obtain the velocity information based on the motion of small tracer
particles seeded in the flow - particle image velocimetry (PIV) measurement technique has turn
into one of the most widespread technique in fluid mechanics experimental research. Particle image
velocimetry is a powerful tool in quantitative flow analysis (Raffel et al., 1998). In its basic configu-
ration, the technique yields two velocity components normal to the viewing direction within a thin
light sheet placed at proper location and orientation within the flow. PIV setups may be classified
according to the number of dimensions involved or more precisely by the number of velocity com-
ponents (C) and the dimensions (D) of the flow volume inspected (Hinsch, 1995). Thus the original
version is termed a 2C2D-technique and described in the attached appendix C.
Fluid velocity, as the characteristic parameter of the flow is, by definition, a vector quantity, and
requires for the complete specification three components u, v and w in the x, y and z directions,
respectively. In addition, many flow features of interest have the three-dimensional nature. For
example, the structural characteristics of flows are revealed only by their spatial features. Hence,
the ultimate goal of the experimental fluid mechanics is the experimental tool that will be capable
to measure the three-component velocity vector of flows in different conditions, with the full four-
dimensionality (i.e., three-dimensional and with high time resolution).
In recent times, much effort applied to the development of flow velocimetry in several dimensions.
There are some innovative techniques that have been presently attempted to solve the problem of
the three-dimensional flow velocity measurements.
2.3.1 PIV-based techniques
The need for the instantaneous investigation of flow fields in three dimensions has inspired several
approaches to extend the well-established concepts of particle velocimetry beyond the plane-wise
recording of two-dimensional transversal velocity vectors (Hinsch, 1995). The third component in

the sheet volume (3C2D) has been tackled by a variety of approaches. At present, the most promising
method is to combine two traditional PIV recordings at different angles to a stereo image yielding
the out-of-plane velocity component.
Recent advances in PIV technique have been directed toward obtaining the all three-components
of fluid velocity vectors in a plane or in a volume simultaneously to allow the application of PIV
technique to more complex flow phenomena. Several three-dimensional PIV methods or techniques
had been developed successfully in the recent years, which include Holographic PIV (HPIV) method
(see for example Barnhart et al. (1994), and Zhang et al. (1997)), three-dimensional Particle-Tracking
Velocimetry (3D-PTV) method (Nishio et al., 1989) and Stereoscopic PIV (SPIV) method that will
be discussed in the present study.
HPIV technique, which utilizes holography technique to do PIV recording, enables the measure-
ment of three components of velocity vectors throughout a volume of fluid flow with highest (between
existing PIV methods) measurement precision and spatial resolution. However, HPIV is also the
most complex, requires a significant investment in equipment and the development of advanced data
processing techniques. The most significant drawback of this advanced technique is its ”non-digital”
nature. The time intervals of the recording, reconstruction and processing steps are too long to be
useful for the accepted statistical approaches in the turbulence research.
Three-dimensional PTV technique uses three (or more) cameras to record the positions of the
tracer particles in a measurement volume from three different view directions (see for example,
Heinrich, 1999) and, through three-dimensional image reconstruction, the locations of the tracer
particles in the measurement volume are determined. By using particle-tracking operation, the three
dimensional displacements of the tracer particles in the measurement volume could be calculated.
However, the small-scale vortices and turbulent structures in the flow field theoretically can not be
identified successfully from the 3-D PTV results due to its low spatial resolution.
Stereoscopic PIV technique is a most straightforward (but not always easy accomplished) method
for the velocity three components measurement in the illuminating laser sheet plane. It uses two
cameras at different view axis or offset distance to do stereoscopic image recording. By doing the
view reconciliation, the corresponding image segments in the two views are matched to get three
components of the flow velocity vectors. Compared with 3-D PTV method mentioned above, the

stereoscopic PIV measurement results have much higher spatial resolution. However, the conven-
tional stereoscopic PIV measurement results within one single plane often yields not enough infor-
mation to answer the fluid governing equations (such as Navier-Stokes equations) that summarized
our fluid-mechanical knowledge. In the meanwhile, for most of the turbulent flows like turbulent
boundary layer flow, vorticity vector (three-component) field is another very important quantity
to evaluate the evolution and interaction of the vortices and the coherent structures in the vortex
flows besides the velocity vector. In the statistical theory of turbulence, the spatial and temporal
correlation terms of the fluid variables like velocity together with the spectrum of the fluctuations
are very important for the development of turbulence models. Such information about the fluid
flows obviously can not be obtained from the conventional stereoscopic PIV measurement results,
which were obtained at one single plane of the objective fluid flow. The detailed description of the
SPIV basic principles is provided in the Appendix D.
2.3.2 Hot-Foil Infrared Imaging Technique (HFIR)
The hot-foil technique is basically a thermal visualization method, which has been developed by
Hetsroni and co-workers (Hetsroni et al., 1996). This non-intrusive measurement method makes use
of an infrared (IR) thermography of a heated foil located at the bound wall and has been applied
to explore various thermal and hydrodynamic problems, e.g. Hetsroni et al. (1997) (and references
therein). The flow velocity at the most near-wall region affects the heat convection process and,
therefore the temperature field on the foil surface. The temperature field is remotely measured
from the other side of the foil by an infrared visualization. Captured two-dimensional images of the
temperature field provide indirect visualization of the velocity field at the wall. The low- and high-
temperature thermal streaks are clearly visualized by the technique, as presented by Hetsroni et al.
(1996). The basic assumption, which has been proved by numerous measurements, is, that thermal
patterns observed by the hot-foil infrared imaging technique, are footprints of the low- and high-
speed velocity streaks in the near-wall region. Moreover, recent numerical simulation research of Li
(2000) had shown the connection between thermal and velocity streaks in the turbulent boundary
layer flow. Several imaging processing approaches have been developed to characterize the structural
parameters of the thermal patterns by Zacksenhouse et al. (2001).

2.4 Coherent structure identification
None of these methods (streamlines, vorticity lines, vorticity magnitudes, complex eigen-
values of the deformation-rate tensor, and elongated regions of low pressure) are based on
criteria that are both necessary and sufficient for the presence of a vortex, and the debate
over vortex detection techniques continues in parallel with the debate over a definition
of a vortex.
(Robinson, 1991a).
2.4.1 Introduction
It seems that the study of the eddy structure of turbulence is the most fruitful direction for turbu-
lence research at present. It follows that the central activity of turbulence research should now be
(or needed to be) centered on the measurements, analysis, and conceptual understanding of eddy
structures, for different classes of turbulent flow. Because eddy structure cannot be universal, this
research is mainly concerned with structures in one class of turbulence, namely boundary layer flow,
although some of approaches presented here may have application in other types of turbulent flow.
Finally, it should be remembered that research into turbulent structure is providing concepts
that lead to modifications and new designs of turbulent flow to solve engineering and environmental
problems. For example, it is well known that devices for drag reduction and for improvements in
heat transfer, combustion, noise suppression, etc., have been based on the results of recent research
in this field.
Two principal portions of the description process and understanding the physics of turbulent
boundary layer flow are
Identification - a robust, unambiguous technique for distinguishing coherent flow structures,
Significance - the coherency analysis is based on the ensemble data analysis, (conditional or un-
conditional), and only this way it becomes statistically significant.
Herein presented several identification and decomposition techniques used by most of the exper-
imental and numerical simulation researches.

2.4.2 Discriminant-based structure identification
This methodology is based on the use of invariants of the velocity gradient tensor and related scalar
quantity, the cubic discriminant. The technique treats each point in a flow field as a critical point
as seen by an observer moving with the fluid particle at the point and at the instant of time in
question (see Chong et al., 1990, for details). The approach considers fluid motions describable by
the Taylor series expansion of the velocity vector u(x0, t) in a small neighborhood h of any point x0
(i.e. x = x0 + h) in R3
domain:
ũ(x0, t) = ũ(x, t) + Aij · h + O(h2
), (2.1)
where Aij is the velocity gradient tensor at the point x0:
Aij = ∇ũ(x, t)|x0
=






∂u
∂x
∂u
∂y
∂u
∂z
∂v
∂x
∂v
∂y
∂v
∂z
∂w
∂x
∂w
∂y
∂w
∂z






(2.2)
The shape of the solution trajectories of the fluid motion velocity equations, (in other words, in-
stantaneous streamlines) can be classified according to the eigenvalues and eigenvectors of the Aij.
The eigenvalues are the roots of the characteristic cubic equation for this tensor, given by
λ3
+ Pλ2
+ Qλ + R = 0 (2.3)
The coefficients of this polynomial are the invariants of the velocity gradient tensor given by
P = −Aij , (2.4)
Q =
1
2
P2
−
1
2
AikAki , (2.5)
R = −
1
3
P3
+ PQ −
1
3
AikAknAni . (2.6)
The first invariant P is identically zero for incompressible flow, so the nature of the roots of equation
(2.3) is determined by the sign of the discriminant of Aij, defined as:
D =
27
4
R2
+ Q3
. (2.7)
Using the described above quantities, the local geometry of three-dimensional instantaneous
streamlines around any point in a turbulent flow field can be categorized using the invariants Q

and R and the discriminant D. An important feature of this method is that both invariants, and
consequently the discriminant are invariant under any affine transformation (i.e. invariant under
non-uniform translations and independent of the orientation of the coordinate system).
The second invariant Q can be broken into two terms
Q =
1
2
(ΩijΩij − SijSij) (2.8)
where Ωij = 1
2 A − AT

is the antisymmetric, rate-of-rotation tensor and Sij = 1
2 A + AT

is the
symmetric rate-of-strain tensor. This expression highlights the fact that the local flow pattern is
determined by a tradeoff between rotation and strain.
The invariants of the velocity gradient, rate-of-rotation and rate-of-strain tensors were used,
for instance, by Blackburn et al. (1996). The isocontours of D ≈ 0, observed by authors, have
provided the picture of ’horseshoe’ vortices, previously proposed by a number of investigators since
Theodorsen (1952), including Townsend (1956), Head and Bandyopadhyay (1981) and others.
2.4.3 Vortex-induced pressure minimum identification
The method for vortex identification proposed by Jeong and Hussain (1995) captures the pressure
minima in planes perpendicular to the vortex axis, based on the connection between the tensor
S2
+ Ω2
and the Hessian of the pressure Hp
S2
+ Ω2
= −
1
ρ
Hp (2.9)
under assumption of neglected unsteady irrotational straining. This equation shows the connection
between local stretching and rotation and the pressure field p. The Hessian H is defined by
Hp =






∂2
p
∂x2
∂2
p
∂x∂y
∂2
p
∂x∂z
∂2
p
∂y∂x
∂2
p
∂y2
∂2
p
∂y∂z
∂2
p
∂z∂x
∂2
p
∂z∂y
∂2
p
∂z2






(2.10)
If the pressure has a local minima, its Hessian must be positive definite. Thus, the the tensor
S2
+ Ω2
should be negative definite, and this condition satisfied when two negative eigenvalues
occur. Therefore, if the second largest eigenvalue λ2 0 (λ1 λ2 λ3), it means two negative
eigenvalues, and a local pressure minimum.

2.4.4 Statistical expansion of the velocity field: the proper orthogonal
decomposition
The proper orthogonal decomposition (POD) as it used in the current research, was introduced in
general by the textbook by Hinze (1975), where the results of the original authors, Bakewell and
Lumley (1967) and Lumley (1967) are quoted as follows:
With a novel1
orthogonal decomposition of the u1-velocity component, proposed by Lumley
(1967) mad an attempt to obtain some information concerning the large-eddy structure close
to the wall. These large eddies were suggested to exist by Townsend (1956), was identified by
Bakewell and Lumley (1967) with the most energetic eigenfunctions in the decomposition. The
size of the eddies thus corresponds roughly with the size of the energy-containing eddies. They
concluded that the structure must consist of a pair of contra-rotating streamwise vortices with a
strongly concentrated ejection from the wall, creating in this way a defect in the U1-distribution
at some distance from the wall. The structure is similar to the picture given by Townsend (1956)
of ”attached” eddies elongated in the streamwise direction. The position of the centers of these
eddies was estimated to be roughly at x+
2 ≈ 50, while the spanwise spacing was roughly λ+
3 ≈ 80,
which is of the same order of magnitude as has been concluded from direct visual observation
studies (Kline et al., 1967). In a later publication (Lumley, 1971) proposed a slightly different
definition, namely that the large eddy corresponds with the motion which can most efficiently
extract energy from the mean motion, and loses as little as possible energy through dissipation.
The proper orthogonal decomposition (POD) or Karhunen-Loève expansion is a classical tool
of probability theory. Lumley (1970) introduced it in the field of hydrodynamics at a time when
a need for mathematical definition of coherent structures in turbulence raised up. Generally, the
POD theorem of probability theory states that a random function can be expanded as a series of
deterministic functions with random coefficients, so that is possible to separate the deterministic
part from the random one. Each scalar function can be decomposed into orthogonal deterministic
functions φj(x) (i.e. POD modes) and random coefficients aj in the following manner
u(x) =
∞
X
j=1
ajφj(x) . (2.11)
11975

The optimal basis, i.e., the most efficient type of the φj(x) functions, which are maximize the
averaged projection of u onto φ, are eigenfunctions of the integral equation (see Holmes et al., 1996,
for the prove)
1
Z
0
hu(x)u∗
(x0
)i φ(x0
) dx0
= λφ(x) . (2.12)
whose kernel is the two-point correlation function (or auto-correlation function)
hu(x)u∗
(x0
)i ≡ R(x, x0
)
where h·i denotes the averaging operation.
It is well-known that the energy of the stochastic signal is given by the sum of the eigenvalues
so that each eigenvalue taken individually represents the energy contribution of the corresponding
term in equation (2.11). Lumley (1970) performed the generalization to vectorial functions to ex-
tract velocity structures from turbulent flows, decomposing the velocity field as a spatial vectorial
function, the most energetic (spatial) eigenfunctions representing the ’eddies’ of the flow. Although
the technique is criticized due to its empirical nature, i.e. the need to extract data from enormous
databases, it has the optimal convergence speed advantage over any other extraction technique. In
addition, despite the fact that we shall almost exclusively apply the POD to non-linear problems, it
is a absolutely linear procedure, and the nested sequence of subspaces are linear, even if the source
of the data is non-linear. Linearity is the source of the method’s strengths as well as its limitations,
as pointed out by Holmes et al. (1996).
2.4.5 Practical implementation of POD
Lumley (1970) refers to these eigenfunctions as coherent structures of the data. Whether
or not they would appear as spatial structures in a laboratory experiment is questionable.
Nevertheless, there is cause to believe that they will be present at least indirectly. Perhaps
an actual structure will consist of a linear combination of eigenfunctions. Sirovich
(1987b)

2.4.6 Calculations
Lumley (1970) introduced the Karhunen-Loéve decomposition method of the random functions to
the turbulent flow research to use it as an unbiased method for discrete data set, such as experimental
or numerical data. It is known that in the continuous case the probability density function (PDF)
provides the full description of the of continuous random functions. The integral of the PDF defines
the mean value of the random vector, and the distribution of the random vectors around the mean
is determined by using the covariance matrix. The optimal presentation of the random set defined
above is based on the eigenvectors and eigenvalues of the the covariance matrix.
In the discrete case (such as PIV or DNS data) the flow quantities are presented as the set
of (random) vectors that approves the second order statistical property – the existence of optimal
representation by eigenfunctions. If the set of M vectors of length n is presented as:
{ui}
M
i=1 , ui = [u1, u2, . . . , un]T
(2.13)
then the discrete approximation of the autocorrelation kernel R is known as the covariance matrix:
C =
1
M
M
X
i=1
ui · uT
i (2.14)
Herein we assume the the data is the field of fluctuations, treated as random data. If the data
analysis is of the instantaneous flow quantity (such as instantaneous velocity or vorticity, for in-
stance), then first the statistical average (denoted by¯or by h·i, interchangeably) is calculated by
the approximation:
ū =
1
M
M
X
i=1
ui (2.15)
and then is subtracted from the data vector set:
ui = ui − ū (2.16)
Then the analysis is done by using the fluctuating field, similar to the spatio-temporal data analysis
performed by Heiland (1992). We point out that the covariance matrix is an N × N matrix, where
N is the spatial resolution of a vector (e.g., for the PIV data it is the total number of the vectors
within the flow field). For large N (e.g., N = 1000 vectors for the usual PIV analysis), the covariance
matrix becomes too large for massive computation. In practice, most of the POD analysis, shown

in the literature, is performed by using the method of snapshots, as described by Sirovich (1987a):
Cij = hui · uji , i, j = 1, . . . , M (2.17)
The matrix Cij is of size M × M, instead of N × N covariance matrix C and in all the cases when
M N (e.g., in PIV analysis the number of realizations will be of order O(10) − O(100)), one
can solve the eigenvalue problem more easily. The symmetry property of the covariance matrix
defines that eigenvalues λi, are nonnegative and its eigenvectors ψi , i = 1, . . . , M form a complete
orthogonal basis (Strang, 1976). The orthogonal eigenfunctions or proper orthogonal modes are
defined by:
φ(n)
=
M
X
i=1
ψ
(n)
i ui , k = 1, . . . , M (2.18)
where ψ
(n)
i is the i-th component of the n-th eigenvector. The original data might be represented2
by using the eigenfunctions φ(n)
and the coefficients an as follows:
ui =
M
X
n=1
anφ
(n)
i (x) (2.19)
in the optimal sense (i.e., by minimizing the L2, the least-squares norm of the error), where the
coefficients are computed from the projection of the data vector onto an eigenfunction:
an =

ui · φ(n)
φ(n) · φ(n)

(2.20)
These coefficients are random and uncorrelated square roots of the eigenvalues:
hanami =





0 n 6= m
λn n = m
(2.21)
By using the property of the orthogonal decomposition, one can show that the ”energy” of the data,
defined as huT
i uii could be calculated by the sum of the eigenvalues:
E =
M
X
i=1
λi (2.22)
and the ”energy fraction” of n-th POD mode (i.e., orthogonal function) is defined as the percentage
of the energy, based on the n-th eigenvalue:
En = λn
, M
X
i=1
λi (2.23)
2”Representation problem” is the header of the section in Lumley (1970)

The decomposition conserves the information, thus any one of the vectors from the original data set
is reconstructed by using the linear combination of all orthogonal modes:
ûi = ū +
M
X
n=1
anφ
(n)
i (2.24)
The low-order approximation of the data is achieved through the reconstruction with finite, small
number of modes (i.e., up to order K M):
ui ≈ ū +
K
X
n=1
anφ
(n)
i (2.25)
2.5 Conditional sampling techniques
Coherent structures such as bursting phenomena near the wall were discovered not by using probe
measurements (e.g. hot-wire), but by flow visualization (see Runstadler et al., 1963). Although
flow visualization allows one to observe coherent motions characterizing turbulence qualitatively,
attempts to measure them with hot-film and laser-Doppler anemometers require the development of
”conditional sampling” techniques. Otherwise, the conventional long-term averaging processes may
not reveal the short-term coherent parts of the velocity fluctuations (Antonia, 1981).
The conditional sampling technique has been extensively used to recognize and yield phase-
or ensemble-averaged information related to organized coherent structures, from both visual data
and velocity fluctuation signals. In order to detect coherent motions from measurements of velocity
fluctuations, one must first know the basic features of the coherent structures from flow visualizations,
only then one could determine a procedure such that only certain significant information is observed.
In general, a conditional sampling of an arbitrary signal q(x1, y1, z1, t + ∆t) and its averaging, as
obtained from the sampling probe, can be defined as
hq(∆x, ∆y, ∆z, ∆t)i =
R
T
q(x1, y1, z1, t + ∆t) · I(x0, y0, z0, t) dt
R
T
I(x0, y0, z0, t) dt
(2.26)
when the detection probe is placed at the spatial point (x0, y0, z0), the sampling probe at the position
(x1, y1, z1, ), such that the spatial lag is defined as ∆x = x1 −x0, ∆y = y1 −y0, ∆z = z1 −z0, and the
time lag as ∆t. The parameter T in the integral denotes the pre-determined averaging time, and the
function I(x0, y0, z0, t) selects the coherent motion in question as it occurs at the point and time. An

appropriate detection function I for coherent motions is not yet well established, although several
detection functions have been proposed by various researches (see for example Nezu and Nakagawa,
1993). The greatest difficulties in establishing the detection functions stem from their random
character, e.g., coherent motions such as bursting events occur randomly in space and time, and
their three-dimensional geometry and convection velocity exhibit a large amount of jitter. However,
these inherent difficulties not necessarily have an unachievable solution. In following sections we will
review the most popular and established techniques. Additional problem is to find the best alignment
phase point between individual events. This cannot be made at exactly the same reference point
for each individual event, due to background turbulence, scale jitter and differences in ages of the
detected events. It has been shown by several investigators (e.g. Blackwelder and Kaplan, 1976)
that such misalignment can result in significant distortion of the deduced pattern of coherent events.
For instance, Yuan and Mokhtarzadeh-Dehghan (1999) used the iterative procedure of searching
the maximum value of the correlation between each individual event and the event ensemble and
shifting the ensemble to that point before following iteration.
2.5.1 u − v quadrant technique
The instantaneous Reynolds stress signal uv is used to detect basic features of bursting motions
because it is directly related to bursting phenomena. Since ejection and sweep motions contribute
the most to turbulence production, sorting functions Ie(t) for ejections and Is(t) for sweeps are
defined as follows:
Ie(t) =





1 u 0, v 0
0 otherwise
, (2.27)
Is(t) =





1 u 0, v 0
0 otherwise
. (2.28)
(2.29)
Unfortunately, the u − v quadrant sorting functions Ie(t) and Is(t) cannot be used directly as
a detection function for bursting motions because of interaction motions. A threshold level H is
introduced and assumed that ejection or sweep motions occur only if |u(t)v(t)| ≤ Hurmsvrms by Lu
and Willmarth (1973). The determination of a threshold level H is, however, more or less arbitrary.

2.5.2 Variable-interval-time-average technique
Blackwelder and Kaplan (1976) developed a variable-interval-time-average (VITA) of velocity fluc-
tuations u(t) as follows:
û(t, T) =
1
T
t+T/w
Z
t−T/2
u(t) dt (2.30)
Then the detection function has been defined as:
I(t) =





1
(u(t)2) − (û(t, T))
2
k · u2
rms
0 otherwise
(2.31)
where T is a short averaging time, of about the same size as a time scale of the bursting motions, and
k is a threshold level. In signal processing the equation 2.30 is known as a low-pass filter expression,
therefore, the value of the detection function is a band-pass-filter signal and it forms a localized
measure of turbulent energy. The VITA technique detects the intermittently generated turbulence,
which corresponds to the transition from ejection to sweep motions and vice versa.
Since the space and time scales of individual bursting motions vary rather randomly, some events
could not be detected due to the phase jitter of bursting motions.
2.5.3 Variable Interval Space Averaging (VISA)
We propose to adopt the VITA technique to use with PIV experimental data, i.e., velocity fields
with high spatial and very low temporal resolution. Clearly that we have to transform the method
developed in the time domain to the space domain. Therefore, the variable-interval time-average
will be replaced by the variable-interval space-average (VISA). The definition will be given in one
dimension for the simplicity, but it is easily extrapolated to two and three dimensions. The main
problem is the implementation: the size of the space window in two dimensions has to be modified
in two dimensions and therefore extends very significantly the computation time.
û(x, L) =
1
L
x+L/2
Z
x−L/2
u(x) dx (2.32)
and the detection function is:
I(x) =





1
(u(x)2) − (û(x, L))
2
k · u2
rms
0 otherwise
(2.33)

In 2D case, this identification method equations have the view of:
û(xi, xj, Li, Lj) =
1
LiLj
xi+Li/2
Z
xi−Li/2
xj +Lj /2
Z
xj −Lj /2
u(xi, xj) dxi dxj (2.34)
and the detection function is:
I(xi, xj) =





1
(u(xi, xj)2) − (û(xi, xj, Li, Lj))
2
k · u2
rms
0 otherwise
(2.35)
The proposed VISA technique could be applied to any signal of the flow, and if we apply this
conditional sampling on vorticity field we just replicate the vorticity based identification technique,
proposed by Hayakawa (1992) and summarized in the following section 2.5.4
2.5.4 Vorticity based identification
This type of identification is build on the assumption that coherent structures in fluid turbulence
should be characterized by coherent vorticity – the underlying instantaneously space-correlated vor-
ticity – and that vortex dynamics is a way for understanding the dynamics of coherent structures,
their role in turbulent transport phenomena (namely entrainment, mixing, heat transfer, chemical
reaction, and generation of drag and aerodynamics noise).
While the discovery of spatially coherent, recurrent, large-scale flow events, often called ”coherent
structures” in turbulent shear flows has been owing to flow visualization studies, one has to rely
upon a certain quantitative method in order to gain a further insight into the detailed characteristics
and dynamical roles of these structures.
The technique of conditional sampling and averaging has been one of the most widely used
methods in laboratory experiments. The general descriptions and historical overview of the technique
are provided by Van Atta (1974) and Antonia (1981), among others.
What one obtains from the conditional sampling is a ”conditional average”, which is defined as
an ensemble average taken over many events that satisfy a certain, prescribed condition . Implicit
working rules in obtaining the conditional average are:
1. the existence of identifiable, recurrent flow events,
2. the selection of similar events through conditioning,

3. the decomposition of any sampling signal f into an ensemble-averaged (i.e. ”coherent”) part
hfi and the remained part fr through the averaging; f = hfi + fr.
In spite of its broad use, the conditional sampling involves two major problems. One is concerned
with ”subjectivity”, which enters in the process of conditioning, i.e., in the choice of proper condi-
tioning signals and the decision of suitable conditions. In general, the ensemble average hfi more
or less depends on the detection scheme used (see e.g., Yuan and Mokhtarzadeh-Dehghan (1999)).
The other problem is the so-called ”jitter”, which occurs in the process of sampling/averaging. This
is caused by the fact that a signal sampling point and a detection (or ”trigger”) point are usually
different, both in space and time. Consequently various random factors of individual flow events
enter into the averaging process and lead to the loss of phase information, eventually causing a
large degradation of the ensemble-averaged result. Since variations of individual events in their
detail, movement and history arise from the inherent nature of turbulence, any conditional sampling
technique cannot be entirely free from the jitter problem.
2.5.5 Detection: Conditioning
In the present technique, the detection conditions are imposed on the ”strength” and ”size” of
smoothed vorticity concentrations, ω̃. The strength is discriminated by applying a threshold Th1
to
ω̃
ω̃ Th1
, Th1
= k1SM , SM =

∂ū
∂y

max
. (2.36)
Here, the local maximum mean shear SM is used for fixing the threshold level. The ”middle
point” of the event of ω̃ being higher than Th1
is assigned to a trigger instant3
. The condition is
that ω̃ values around a triggered point are simultaneously greater than another threshold Th2
ω̃(xc ± ∆x, yc ± ∆y) Th2
(2.37)
Here ∆x and ∆y could be nearly equal to choose nearly circular structures, or in contrast,
significantly different to choose also elongated, elliptical structures.
3OK, here it is: for x-y (and y-z) plane we should average all vorticity events for the same y positions, or somehow
limit it to these positions. For x-z plane it is probably possible to average all events at all z and x positions, without
any difference. April 2002

2.5.6 Eduction: Ensemble average
Large scale events are accepted only when all the criteria above are satisfied, so that weaker, shifted,
smaller scale or highly distorted events are discarded. Once the trigger instants are determined, the
accepted realizations are relatively aligned with respect to each center, and ensemble averages of
velocity components, huii, are calculated. It is important to note that the ensemble averages are
computed from original, unsmoothed velocity signals; the smoothed signals have been used only as
a means of selecting similar large-scale events.
2.5.7 Realignment: Signal Enhancement
In order to align of individual structure centers, we take the cross-correlation between the vorticity-
signal segments of each realization and the initial ensemble average hωi. Each realization is then
relatively shifted by the time delay of the peak correlation.
2.5.8 Advantages
Major advantageous features of the present technique are summarized below:
1. The whole procedure involved in the technique relies on an intrinsic flow property (i.e., vor-
ticity) of organized turbulence structures, and hence it is conceptually self-consistent and
objective, provided that those events are presumed to be characterized by spatially correlated
vorticity.
2. The presence of large-scale vortical events is recognized with filtered vorticity maps, which
give a physical perception of instantaneous fields, more quantitatively than that from flow
visualization pictures.
2.5.9 Pattern recognition techniques
In order to avoid the phase jitter, Wallace et al. (1977) proposed a pattern recognition technique.
In one cycle of bursting motion is defined as a duration from the detection of one typical phase
until the next occurrence, an ensemble-averaged pattern is recognized clearly if the time sequence of
bursting motions is normalized by its individual duration. The authors (Wallace et al., 1977) defined

a typical phase in terms of maximum value of ∂u/∂t because they observed that transition from
ejection to sweep motions occurs more rapidly. It was shown that pattern recognition technique
provides similar results to those of quadrant or VITA techniques. However, the main disadvantage
of this technique, is that it cannot, in principle, serve for the any kind of ”real-time” analysis of
space-time structures of coherent motions .
Ferre and Giralt (1989) introduced the improved pattern-recognition procedure, originally de-
veloped by Mumford (1982) and which uses a velocity model to check the structural characteristics
of the flow. The proposed large-scale motion detector prepared to exhibit at least three important
properties: (i) not distorted by background, fine-scale turbulence, (ii) amplitude independent, i.e.
prepared to detect motions with low energy content, and (iii) size independent (it is important to
note that the real limitation of the size-independent condition is the maximum number of anemome-
ters that could be used in the experiments at that time). The procedure is based on the introduction
of the initial estimate of large-scale motion from the previous knowledge, search for ’similar’ pat-
terns in the flow measured data base (’similarity’ or ’closeness’ is introduced as an Euclidean or
other distance measure), finding the best alignment position by cross-correlation plane analysis, and
using the resulted pattern and an initial estimate for the next iteration. This iterative process first
described by Mumford (1982), ensured that even in the case of a bad initial estimate, the original
features of the data can be extracted. Authors (Ferre and Giralt, 1989) extracted the large-scale
eddies with the presented technique in the near wake behind a cylinder, using two-dimensional ve-
locity maps provided by hot-wire array. The analysis was extended to the three-dimensional data
analysis by Ferre et al. (1990), but yet implemented using the two-dimensional flow velocity maps
from the hot-wire anemometers array.
2.5.10 Comparison between different conditional sampling techniques
Yuan and Mokhtarzadeh-Dehghan (1994, 1999) compared 12 different conditional sampling methods
on a one-to-one basis and presented the comprehensive degree of correspondence investigation. It
has been shown, however, that no two methods detect exactly the same event ensemble. Some
methods provided good correspondence (about 70%), while others have identified even ensembles,
which bear little correspondence to those detected by other methods. The conditionally averaged
patterns of hui, hvi, and huvi by various methods appeared to be very different in some cases. The

reasons proposed by authors (Yuan and Mokhtarzadeh-Dehghan, 1994) included phase jitter and
sensitivity of different techniques to different phases of the bursting process. A synthesized pattern
for a complete bursting process consisting of an ejection followed by a sweep was suggested by
combining different phases of the event from different conditional-sampling methods.
2.5.11 Recent work
Kline and Portela (1997) proposed the following definition of the vortex: ”a swirling motion around
a ’nucleus’ viewed from a reference frame attached to the swirling motion”. The essence of the
definition is as following: (i) vortices are volumes in the flow, and thus never the same as lines of
vorticity (which are not volumes, clearly), and (ii) typical vortices contain a vortex tube, but the
opposite is not true (i.e., many vortex tubes are not vortices). Following the proposed definition,
the authors attempted to find vortices in two-dimensional planes of DNS results. The shown picture
is very similar to findings of Robinson (1991a) who used the same DNS database and low-pressure
threshold to extract vortical structures.
Tomkins et al. (1998) made use of the hairpin vortex model to define vortex passage signatures
in a spanwise-wall normal (y − z) and a streamwise-wall normal (x − y) planes. Using the defined
hairpin vortex signatures and quadrant sampling technique, authors found packets of hairpin vortices
at the flow velocity fields of measured by PIV and numerically simulated turbulent boundary layer.
Piomelli et al. (1993) numerically seeded the sublayer of a well resolved turbulent channel flow
LES with ”massless” particles which were found to form low-speed streaks. Ejections of particles
from these streaks spatially coincided with compact regions of high Q2 Reynolds shear stress. The
Q2 and ejected particle regions occurred either between counter-rotating quasi-streamwise vortices
or on the upwelling side of single vortices. The vortices had angles of inclination to the wall of about
8◦
.

Chapter 3
Mathematical background
3.1 Basic definitions
Here we recall the notation of the flow variables: tilde (
˜
) denotes the instantaneous value the mean
value (interchangeably time and ensemble average) and small letters for fluctuations. For example
the instantaneous velocity ũi is decomposed into a mean flow Ui and velocity fluctuations ui, such
that
ũi = Ui + ui (3.1)
The mean flow velocity is a time average defined by
Ui = lim
T →∞
1
T
t0+T
Z
t0
ũidt (3.2)
and in discrete case (like PIV or DNS data over discrete grid) is replaced by an ensemble average,
h·i. In addition, the mean values of fluctuations will be denoted by bar¯
:
ui = huii =
1
N
N
X
n=1
{ui}n (3.3)
For a time or ensemble average to make sense, the integrals in (3.2) or sum in (3.3) have to be
independent of time, or in other words the flow has to be steady1
:
∂Ui
∂t
=
∆Ui
∆t
= 0 (3.4)
1We will make use of this relation in Appendix B to prove the steadiness of the PIV experiments.
26

CHAPTER 3. MATHEMATICAL BACKGROUND 27
The instantaneous equations of motion of an incompressible fluid are
∂ũi
∂t
+ ũj
∂ũi
∂xj
=
1
ρ
∂σij
∂xj
, (3.5)
∂ũi
∂xi
= 0, (3.6)
where σ̃ij is the stress tensor, and for Newtonian fluids is defined by
σ̃ij = −p̃δij + 2µs̃ij (3.7)
where δij is the Kronecker delta, p̃ is the pressure, µ is the dynamic viscosity and rate of strain s̃ij
is given by
s̃ij =
1
2

∂ũi
∂xj
+
∂ ˜
uj
∂xi

(3.8)
The stress σ̃ij is also decomposed into mean and fluctuating components, such that
p̃ = P + p (3.9)
Sij =
1
2

∂Ui
∂xj
+
∂Uj
∂xi

, sij =
1
2

∂ui
∂xj
+
∂uj
∂xi

(3.10)
3.1.1 Correlations
Herein we define the correlation between two variables as the average of their product, and by using
the above defined decomposition and averaging operator we can show that:
ũiũj = UiUj + uiuj. (3.11)
Thus we show that two variables are defined as uncorrelated if their fluctuating values provides that
uiuj = 0. We can normalize the correlation by dividing this term by the square root of the product
of variances u2
i , and the resulting correlation coefficient is:
cij ≡
uiuj

u2
i · u2
j
1/2
(3.12)
We will use the correlation coefficient (and its direct analogy, correlation tensor) and root-mean-
square (the square root of the variance) quantities in our post-analysis and identification algorithms.

3.1.2 Mean flow equations and Reynolds stress
The equations of motion for the mean flow Ui are obtained by substituting (3.1) into (3.5) and taking
the average of all terms:
Uj
∂Ui
∂xj
+ uj
∂ui
∂xj
=
1
ρ
∂
∂xj
Σij, Σij = −Pδij + 2µSij (3.13)
The continuity equation becomes:
∂ũi
∂xi
=
∂
∂xi
(Ui + ui) ⇒
∂Ui
∂xi
= 0,
∂ui
∂xi
= 0. (3.14)
Substituting this result into the mean flow Navier-Stokes equations (3.13), we recognize that the
term uj
∂ui
∂xj
can be written in the form ∂
∂xj
uiuj, analogous to the convection term (the first term in
Eq. 3.13). This term represents the mean transport of fluctuating momentum by turbulent velocity
fluctuations. We should emphasize, that if ui and uj are uncorrelated according to the correlation
definition, given in (3.11), there would be no turbulent momentum transfer. Moreover, this term
in (3.13) represents the exchange of the momentum between the turbulence (second term) and the
mean flow (first term). Following the Tennekes and Lumley (1972) we can define the total mean
stress Tij in a turbulent flow as follows:
Tij = −Pδij + 2µSij − ρuiuj (3.15)
The important part of our analysis of the turbulent flow in a flume makes use of the second term
of the above equation, the mean rate-of-strain Sij, and the third term, the Reynolds stress tensor,
ρuiuj. It has been recognized for a long time, that the off-diagonal components of the Reynolds
stress tensor (i.e., i 6= j) are shear stresses and play a dominant role in the momentum transfer by
turbulent motion.
3.1.3 Reynolds stress and vortex stretching
This section uses the idea of the textbook by Tennekes and Lumley (1972) and will be used later
in our analysis and characterization method as the another commendation of the results. The need
for turbulent momentum transport in turbulent boundary layer (let us for simplicity discuss here
only x1 − x2 plane) requires that the velocity fluctuations u1 and u2 have to be correlated (i.e.,
non-zero Reynolds stress). On the other hand, the flow near the solid boundary is a shear flow

with ∂U1/∂x2 0, and when u2 is positive (i.e., lifting upward motion), u1 should have negative
values more frequently than positive ones. This logic brings the basic physical idea that ”the energy
of the eddies has to be maintained by the shear flow, because they are continuously losing energy
to smaller eddies” (Tennekes and Lumley, 1972). Thus, the efficiency of the coherent structures
(denoted ’eddies’ in Tennekes and Lumley (1972)) is measured by their ability to absorb energy
from the shear flow. It has been recognized for a long period of time (e.g., Townsend, 1956) that
the most effective eddies, that maintain the correlation between the fluctuating velocity components
and extracts energy from the mean flow, are vortices whose principal axis is roughly aligned
with the principal axis of the mean strain rate, (Tennekes and Lumley, 1972, Figure 2.5).
3.2 Kinetic energy of the mean and turbulent flow
The equation of the kinetic energy of mean flow, UiUi is obtained by multiplying the mean flow
Navier-Stokes equation (3.13) by Ui:
Uj
∂
∂xj

1
2
UiUi

= −
∂
∂xj

P
ρ
Uj + uiujUi − 2νUiSij

− 2νSijSij + uiujSij. (3.16)
In this equation, we should point out two terms (that will be used later in the analysis): viscous
dissipation term, 2νSijSij, and the term that represents the deformation work done by turbulent
stresses and thus serves as an input of the energy, −huiujiSij, known as turbulent energy production.
In the similar manner, if one multiplies the Naiver-Stokes equations by ũi and takes the average
of all terms, the resulting equation of the mean kinetic energy 1
2 uiui is obtained (also known as
turbulent energy budget):
Uj
∂
∂xj

1
2
uiui

= −
∂
∂xj

1
ρ
p uj − 2νuisij +
1
2
uiuiuj

− 2νsijsij − uiujSij. (3.17)
The terms on the right-hand side from left to right are: pressure-gradient work, transport term of
viscous stresses, transport by turbulent velocity fluctuations, viscous deformation work (dissipation),
and deformation work by Reynolds stresses. First three terms, similar to the analogous terms in
the mean flow energy equations, are responsible for the transport of energy if there is an input or
output of the control volume one considers. Thus, if the energy integral over the closed control
volume is zero, these terms will be negligible in their contribution to the energy transport. The
two other term (deformation work terms) are more important and will take a significant role in our

analysis. We should notice that the turbulence production term −uiujSij appears in mean- and
fluctuating-energy equations with opposite signs, since this is the responsible term for the energy
transfer between the mean flow and the turbulence. The another deformation term −2νsijsij will
appear always negative and clearly presents viscous dissipation (i.e., the flow away of energy).
3.3 Vorticity and velocity gradient tensor
”All turbulent flows are characterized by high levels of fluctuating vorticity. This is the
feature that distinguishes turbulence from other random fluid motions, like ocean waves
and atmospheric gravity waves”. Tennekes and Lumley (1972)
Definition: The vorticity is the curl of the velocity vector ũi = (u1, u2, u3):
ω = curl ũ = ∇ × ũ =






∂u3
∂x2
− ∂u2
∂x3
∂u1
∂x3
− ∂u3
∂x1
∂u2
∂x1
− ∂u1
∂x2






(3.18)
This definition shows that vorticity is related to the deformation rate, or in other words, velocity
gradient tensor:
∂ũi
∂xj
= ∇ũ =






∂u1
∂x1
∂u1
∂x2
∂u1
∂x3
∂u2
∂x1
∂u2
∂x2
∂v
∂x3
∂u3
∂x1
∂u3
∂x2
∂
∂x3






. (3.19)
The gradient tensor can be separated into a symmetric and a skew-symmetric or antisymmetric
part:
∂ũi
∂xj
= s̃ij + r̃ij (3.20)
Where the symmetric part is a strain rate:
s̃ij =
1
2

∂ũi
∂xj
+
∂ũj
∂xi

(3.21)
and r̃ij be the antisymmetric part of ∂ũi
∂xj
, called the rate-of-rotation tensor
r̃ij =
1
2

∂ũi
∂xj
−
∂ũj
∂xi

(3.22)

Chapter 4
Analysis approach
Analysis of turbulent flow inevitably invoke a statistical description. Individual eddies
occur randomly in space and time and consists of irregular regions of velocity or vorticity.
Statistical theory is a way to fathom the complexity. Durbin and Pettersson Reif (2001)
4.1 Decomposition of turbulent flows
The literature review (Chapter 2) proposes that exists some kind of duality in the turbulent analysis.
From the one hand, by using the Fourier transform from the time to spectral domain, the theory
shows the universality of the turbulent spectrum and prediction of the turbulent structure. From
the other hand, we must admit a lack of the theory when we look at the physical or spatial domain.
Nevertheless, we have a large amount of experimental (Townsend, 1956, Kline et al., 1967, Kaftori
et al., 1994) and numerical (Kim et al., 1987, Schoppa and Hussain, 2000, among others) evidence
for the presence of coherent spatial structures in turbulent flows. The coherent motions correspond
to the vorticity of the flow, condensed into organized motions, which contain most of the energy and
enstrophy of the flow. The spatial organization of these inherently reduces the predicted nonlinearity
of the flow, and the reduction is even larger due to the symmetry of the structures. These coherent
structures seem to play an important, but not yet well understood role in the transport of the mo-
mentum, mass and heat in the turbulent flow. We will show in the following, that the appropriately
31

CHAPTER 4. ANALYSIS APPROACH 32
chosen orthogonal transform, which allows an orthonormal projection of the flow quantities on a min-
imal number of uncorrelated modes (i.e., POD) will represent turbulent flow dynamics in a better
way than with Fourier modes. Fourier transform requires very large number of modes (frequencies)
to describe the flow in a suitable way, and its basis functions (i.e, trigonometric functions) would be
appropriate to present the flow only as a superposition of periodic waves. In contrary, we depict the
turbulent flow as the superposition of coherent structures, and the Fourier spectrum in this case is
meaningless. It is a very common in many transforms to neglect the effect of the analyzing function
(such as trigonometric functions in Fourier transform, or basic flow pattern in pattern recognition
analysis) on the later interpretation of the transformed field. If such case the structure of the basic
function might being interpreted as characteristic of the field under study. In order to reduce such
misleading, we choose the analyzing function in accordance to the intrinsic structure of the field to
be analyzed, that is orthogonal eigenfunctions of the flow field under analysis. Finally we will study
how the turbulent dynamics transports the identified structures, distorts them, and exchanges the
energy by means of such structures. Orthogonal modes of the POD transform, as they demonstrated
in the review, are certainly best candidates for performing the energy decomposition, and for finding
possible coherent structures that characterize the turbulent boundary layer flow dynamics.
Before discussing the actual application of POD modes to the analysis of the turbulent flow field,
we should accentuate two points:
i) First of all, orthogonal modes are useful as a decomposition tool for the study of turbulent flow
if we want to engage some information about the spatial structure of the flow. In contrary, if we
are interested in its spectrum, POD modes are not helpful, and the Fourier transform should
be used instead.
ii) Secondly, we should always keep in mind that POD transform emphasizes the signal fluctuations,
but is insensitive to constant component of the signal (i.e., the average strength). A common
pitfall in interpreting POD modes coefficients (a
(n)
i , λ(n)
), is to link their strength (presented
by gray level intensity in this work) to the signal’s strength, whereas they actually correspond
to fluctuations (i.e., variance) of the signal.

CHAPTER 4. ANALYSIS APPROACH 33
4.2 Guidelines of the analysis
The guidelines of the analysis approach in this work were determined as follows:
• Data analysis is performed without thresholding, and the same filters are applied to all the
data.
• Data has to be statistically significant in order to characterize the structures that exist during
a period of time.
• Analysis is based on a flow characteristic, which strongly represents turbulence (e.g., vorticity).
Accepting the fact, that coherent structures have a dominant role in the turbulent boundary
layer, one can characterize their properties by using any available flow quantity. The first choice is
the velocity field, which actually contains all the necessary information about the turbulent flow.
However, we will show in Section 6.2 that the velocity data lacks the necessary spatial localization,
used to define the topological characteristics of the coherent structures, due to the lack of Lagrangian
invariance. The turbulent flow quantity that used to identify coherent structures in the present work
is the vorticity orthogonal components: ω1, ω2, ω3.
In the present work we adopt the concept of ‘characteristic eddy‘ Lumley (1970), and similarly
to the reconstruction method presented by Gordeyev and Thomas (2002), we consider the linear
combination of the dominant POD modes as a description of the term ‘large scale structure‘:
ω̂i(x) =
N
X
n=1
λ(n)
φ(n)
(x) i = 1, 2, 3. (4.1)
The procedure in (4.1) fulfills the defined guidelines and allows to represent both qualitative and
quantitative characteristics of the coherent structures.

Chapter 5
Experimental apparatus
5.1 The infrastructure
This research consists the experiments within the horizontal open (i.e., free water-air interface) flume
of 4.9 × 0.32 × 0.1 meter as shown in the schematic diagram in figure 5.1.
The entrance and the following part of the flume (up to 2.8 meter downstream) has been produced
from the glass in order to make flow visualization and PIV measurements possible. All necessary
cautions in the entrance have been made: the eddies and recirculating currents damped with the
narrow slits that constructed within the inlet tank (as presented by dotted lines in the figure 5.1),
the baffles are installed into the pipes portion of the tank, the inlet to the channel is made as a
converging channel in order to prepare a smooth entrance, the pump is isolated from the system by
rubber joints fitted to the intake and discharge pipes. The pump is a 0.75 HP, 60 RPM centrifugal
pump. Flowmeter with 0.5% accuracy level (i.e., 0.5% of the measured flow rate scale), based on
the pressure drop measure between the pump and the tanks, continuously records the flow rate. In
order to make the measurement area long enough and avoid the flow depth drop of at the end of the
flume, the flow restrictors (in the form of array of cylinders) are placed at the outlet portion. The
measurements have been performed with treated and filtered tap water.
34

CHAPTER 5. EXPERIMENTAL APPARATUS 35
6 2
3
2
5
4
1
1 1 1 2
1 0
9 1 3
8 7
FDW
L
RQDO
/ L
FHQV
H
2 QO

Figure 5.1: Facility schematic view
.ieqipd zizyz ly ihnikq xe‘z :5.1 xei‘
5.2 Stereoscopic PIV system
A commercial SPIV system, consists of the following subsystems, has been purchased for the current
research from TSI Inc.:
Illumination Two Nd:YAG lasers of 170 mJ/pulse, 15 Hz, 532 nm, and optical system,
Acquisition Two digital CCD cameras of 1000×1000 pixels,
Synchronization and processing Synchronizer, acquisition and post-processing software (Inc.,
1999a).

Figure 5.2: Illumination scheme in two views: front (left) and top (right)
.(oinin)lr hane (l‘nyn) inciw han :mihan ipya dx‘dd znikq :5.2 xei‘
5.2.1 Illumination
The illumination is produced by two, time sequenced, Nd:YAG lasers (Quanta Ray, Spectra Physics),
each pulsing 170 mJ at 15 Hz. The wavelength of the light is 532 nm with a pulse width ∼ 6-7
nanoseconds. The output beam diameter from the laser around of 8 mm and has linear vertical
polarization. The beams of two laser passes threw the same optical system, shown in the following
figure 5.2, with a short time delay between them, controlled by the synchronizer. As it is shown in
the figure 5.2, the light beam 1 from the laser is navigated by the set of the 45◦
mirrors 2 toward
the light sheet formation optics. The laser beam is transformed into the laser sheets by using the
cylindrical lens 3 , which generates a thick sheet of light ( 5 ) from the collimated laser beam by
expanding the light in one axis only. Than, the laser light is delivered through the spherical lens
4 in order to achieve thin laser light sheet with higher laser intensity (see Appendix C for details).
Laser sheet of a 1 mm thickness was found to carry out the requirements, similar to the common
guidelines in the PIV literature (Adrian, 1991, Raffel et al., 1998).

5.2.2 Seeding
A successful measurement by using PIV is based one of the most significant issues - a good seeding.
It relates to the ”Particle” part of the PIV name and it means that the tracer particles have to be
dispersed into the flow field with appropriate seeding concentration. The requirements for particles
are:
• High scattering property,
• The ability of the particles to follow the instantaneous velocity changes of the fluid.
These requirements place the ultimate limit on the accuracy of the velocity field measure-
ment.Another important characteristics of the seeding are the high spatial concentration and the
size uniformity of the particles. The choice is based on the trade-off between the tracing capabilities
of the seeded particles and the high signal-to-noise (SNR) ratio of the scattered light signal. Accord-
ing to the common practice in PIV measurement, that was comprehensively developed by Melling
(1997), we calculated the optimal size of the particles to be ∼ 10νm. The calculations were based
on the priori experimental data of Kaftori et al. (1998), and approved by later PIV measurements.
The above size is the most appropriate choice for the turbulent boundary layer flow with ≈ 5%
level of the streamwise turbulence intensity u0
1, and the Kolmogorov time scale of about 0.1 sec.
In experiments we have used the hollow glass spheres type of particles, with density of 1.1 g/cm3
(Potters Ind.), and a mean diameter of davg = 11.7 µm. The particles were tested with Malvern
analyzer and the mean size is found to be between 9 and 16 µm.
5.2.3 Acquisition and calibration
The images of the particles that scatters the laser induced light, have been captured on the CCD
arrays of two digital cameras of 1K×1K pixels spatial resolution and with the frame rate fitted to
the laser pulse rate of 15 frames-per-second (i.e. 15 pairs of images per second, or in other words,
15 velocity maps per second). The cameras have been installed in the angular configuration along
with the Scheimpflug condition. Each camera capture two separate frames synchronized with the
laser pulses and particle displacements is calculated with the cross-correlation technique. The three-
component velocity field is obtained according to the stereoscopic viewing principle, as it is described
in Appendix D, based on two two-component velocity fields from two cameras.

Calibration The stereoscopic PIV system, installed in the angular configuration, has the following
inherent features:
• The Scheimpflug condition causes perspective distortion (e.g., the rectangle image appears as
a trapezoid),
• Three-dimensional position and displacement of the particle converted first to the two dimen-
sional displacement field, and only later transformed into the three-component velocity field.
In order to achieve a high level of accuracy with the angular configuration, the SPIV analysis uses
the calibration as a first acquisition step. The calibration is performed with the specially designed
commercial grid and the calibration procedure based on the mapping algorithm with the purchased
PIVCalib software (Inc., 1999b). The third, out-of-plane component of the velocity is validated
versus the two-dimensional PIV and the LDV measurements, as it is depicted later in Appendix B.
5.2.4 Synchronization and processing
The programmable pulse delay generator has been purchased with the SPIV system in order to allow
the control over two important characteristics:
• Accurate measurement of the laser pulse delay,
• Synchronization of the laser and the image acquisition.
The acquired images have been analyzed by means of the PIV software that determines the
velocity field. We use the purchased software InsightTM
(Inc., 1999a) that performs the PIV analysis
with the following common steps:
a) Each PIV image is divided up into a interrogation areas over the regular square grid (e.g., 32×32,
64×64, or 128×128 pixels),
b) At each grid window the local displacement is calculated based on the cross-correlation by using
the FFT method,
c) Sub-pixel displacement by using two dimensional Gaussian interpolation,
d) Erroneous vectors are removed by means of common PIV filters (e.g., global and local median
filter),

Figure 5.3: Stereoscopic PIV configuration scheme for the x1 − x2 plane experiment.
x1 − x2 xeyina ieqip xear SPIV -d zkxrn ly dnikq :5.3 xei‘
e) Interpolation applied at the positions of the missing data points.
5.3 Experimental conditions
In this work, the measurements in two- and three-dimensional setups were performed in orthogonal
planes relatively to the flume boundaries, namely streamwise - wall normal x1 − x2, streamwise -
spanwise x1 − x3, and wall normal - spanwise x2 − x3 (figures 5.3-5.5).
The experiments were performed under hundreds of different conditions (i.e., locations, flow
rates, flow heights, camera locations, combined with HFIR and with additives, etc.). Here we list
the representative 8 experimental conditions in Table 5.1. The coordinates of the measured planes
are defined relatively to the coordinate system with the origin at the left lower corner of the inlet of
the flume. The velocity field was measured at a distance 2.5 m from the inlet (i.e. x1 = 2.5 m).

Case No. Plane Coordinates [m] Reh Um [m/s] u∗
[m/s]
1 x − y z = 0.15 21000 0.21 0.011
2 x − y z = 0.15 27000 0.27 0.013
3 x − y z = 0.15 45000 0.45 0.022
4 x − y z = 0.15 57000 0.57 0.027
5 x − z y = 0.003 27000 0.24 0.013
6 x − z y = 0.01 27000 0.24 0.013
7 x − z y = 0.025 27000 0.24 0.013
8 y − z x = 2.50 27000 0.24 0.011
Table 5.1: Experimental parameters of the StereoPIV.
.miieqipd i‘pze mixhnxt :5.1 dlah
5.4 XPIV – Multi-plane Stereoscopic Particle Image Velocime-
try
In this section, the three dimensional extension of the stereoscopic PIV method, XPIV - Multiplane
SPIV is presented, along with the optical scheme, basic principles and image processing algorithm.
The quality of the velocity data is evaluated by using the velocity profiles, turbulent intensity and
the continuity equation characteristics.
5.4.1 Introduction
Experimental investigation of turbulent flows requires techniques that allow three dimensional mea-
surements with high spatial and temporal resolutions. PIV appears to be an appropriate basis for
three dimensional velocity measurements, as it is presented in the literature review, Section 2.3. The
technique has only technological limitations to achieve a temporal resolution due to the illumination
source (lasers) and recording media (CCD) frequencies which are available today.
Understanding the drawbacks and advantages of the obtainable measurement systems led to
the development of the multi-plane stereoscopic velocimetry technique, XPIV . The technique ap-
plies the principles of multi-sheet illumination, stereoscopic imaging and particle image defocusing.
The experimental technique implemented with a stereoscopic PIV system (Section 5.2), based on
additional optics and image processing algorithm.
Section 5.5 presents the optical configurations implemented during the research. Image processing

Figure 5.6: Schematic view of the first type of the optical array.
algorithm is described in section 5.6.
5.5 Optical arrangements
During the research work, two optical arrangement schemes are proposed, both allow the same basic
multi-plane illumination principle, but with several modifications. Each one of the schemes has its
own advantages and drawbacks, as it described below.
5.5.1 Variable light intensity scheme
The first optical arrangement is presented in figure 5.6. The laser beam with linear vertical po-
larization passes the spherical lens 1 to produce a focused laser sheet at the area of interest, and
then it is turned up, from its horizontal direction, toward the optical array, using 45◦
high energy
laser mirror 2 . The beam splitting array comprises of four components in the specified order. The
beam passes through the zero order half plate 3 mounted on the rotation mount, which allows to

change the angle of the linear polarization of the output beam. Next is a high energy cube polarizing
beamsplitter 4 with side size of 1.25 cm which transmits s-polarized light and reflects p-polarized
light. Thus, if the laser beam at its entrance is totally s-polarized, the light will be transmitted
almost completely and the output of the array will be only two parallel beams. In our case, we
can control the partition of the laser beam energy between the first (lowest) plane and two other
sheets. Using an appropriate angle of the quarter plate we can achieve the 1/3−2/3 energy splitting
between the reflected and transmitted beams, respectively. Next component is non polarizing cube
beamsplitter 5 of identical size (1.25 cm) with the predefined 50%–50% relation between the trans-
mitted and reflected beams, and after that is located the right-angle prism of 1.25 cm 6 , which is
used as a mirror to approve the identical distances between three beams. Finally, all three parallel
beams passes the cylindrical lens 7 to get three parallel laser sheets ( 8 ). We should note that the
polarization properties of the laser sheets are not important in this scheme, but only their intensity
and alignment characteristics. All the optical components were mounted on the same optical board.
Two beamsplitters and the prism were placed in a slot to maintain their co-alignment and were
attached without gluing (high energy laser beam can damage the glue, resulting in beam aberra-
tions). The presented optical arrangement forms three parallel laser sheets with a known, physically
defined distance of 1.25 cm between them (i.e. the size of cube beamsplitters and the prism) and
with an adjustable intensity. The easy implementation, optical alignment and adjustable intensity
for several planes are the important advantages of the proposed configuration. The main drawback
of this scheme is that cubic beamsplitters are actually glued prisms, and, therefore has very low
optical damage threshold. Thus they are inappropriate for the usage with high laser intensity and
eventually reduces the dynamic range of the PIV images.
The extension of this optical arrangement to produce four, five or more planes is straightforward.
On the other hand, it is impossible to use this splitting idea to achieve parallel planes with variable
distances between them, rather than presented fixed distance scheme. We have chosen the optical
components to be the smallest ones that are available (0.5” = 1.25 cm) from a commercial catalog,
but this is probably the lowest limit of the proposed scheme with a reasonable cost.

Figure 5.7: Schematic view of the second type of the optical array.
.ipyd beqdn zihte‘d zkxrnd ly iznikq xe‘z :5.7 xei‘
5.5.2 Variable inter-plane distance scheme
This optical arrangement was proposed after the first scheme with the fixed distance and it aims to
overcome (i) the fixed distance limitation, and (ii) low optical damage threshold of the beamsplitting
optics. The following scheme (figure 5.7) is proposed to fulfill the demands. This optical scheme is
based on the different type of the beamsplitters - plane beamsplitters( 3 - 5 ) of the 0.5” diameter
and width of 2.5 - 3 mm. The important advantage of these optical units is their high optical
damage threshold, 10 J/cm2
. In this configuration we use the same type of the beamsplitter, with
only one different parameter, the ratio between the transmitted and reflected light, T/R. The first
beamsplitter 3 is of T/R = 67%, the second 4 is of 50% and the last one 5 is 99%, when the
later actually performs like a mirror. Second advantage is that the distance between the light planes
could be different and adjustable by the appropriate location of the optics, but the main drawback
is the alignment of the light beams, and the time consuming parallelization of the final light sheets.
The extension of this optical arrangement to larger number of planes, is also easy to implement,
however, the alignment of the planes should be probably solved by using more flexible optical

mounting scheme.
5.5.3 Calibration of XPIV
The scattering light of the particles from the three laser sheets, produced by one of the optical
schemes, captured by two CCD cameras in the stereoscopic configuration. The focus planes of both
cameras is located at the most far laser plane and the particle images entitled herein focus image
The particle images from the middle and the nearest planes are entitled defocus image 1 and defocus
image 2, respectively. According to the defocus principle, described in the literature review, the size
and the intensity of the particles in these two planes are different and could be roughly estimated by
using the point spread function (PSF) principle. Therefore, the image of each particle is convoluted
with the Gaussian two dimensional shape, thus its size grows with the distance from the focus plane,
and its intensity reduces with the same proportion.
During the calibration stage of the stereoscopic PIV technique, the two-plane grid is located at
each one of the light sheet planes, and the focus of the CCD cameras was adjusted to achieve an
accurate calibration images. The calibration procedure results with the three different grid definition
files, automatically utilized by InsightTM
software to calculate the three-component velocity fields
for three different planes.
5.6 Image processing algorithm
5.6.1 Pre-processing of images
As it is mentioned above, the particle images at the most far plane, on which the cameras were
focused during the experiments, are obviously brighter and smaller than images of the particles
from the lowest and the middle planes.
Figure 5.8 depicts the PIV image of 256×256 pixels, that is about 1/4 of the acquired image. The
figure contains particle images from different planes on the non-uniform illuminated background.
The first image processing operation is to enhance the PIV images by removing the background
non-uniform illumination and adjusting the image contrast (Young et al., 1998). The background
illumination was removed by gray scale morphology operator, ”top-hat”, using circular structure

Original XPIV image
50 100 150 200 250
50
100
150
200
250
Figure 5.8: Original three plane PIV image
element B with the radius of 12 pixels:
J = I − (I ◦ B) = I − ((I B) ⊕ B) (5.1)
where I denotes the operated image, J is the resulted image, ◦ is the gray scale ’opening’ operator,
and ⊕ are erosion, and dilation operators, respectively. For the definitions of the image processing
operations in the present work, see Appendix A. The gray scale morphology operations are usually
faster than their linear filter analogy and were performed by using the Image Processing Toolbox
of Matlab r
(MathWorks Inc.). In addition, the image contrast is adjusted by stretching the gray
level intensity histogram to the lowest and the highest values (i.e. for 8 bit images, 0 and 255,
respectively). The enhanced, preprocessed, image is shown in figure 5.9. The following section
describes the image processing algorithm used to identify particle images as objects in the PIV
image and classify them to one of three groups of particles, according to the illumination plane.

Enhanced XPIV image
50 100 150 200 250
50
100
150
200
250
Figure 5.9: Enhanced three plane PIV image
5.6.2 Particle images in the plane of focus
Particle images originating at the plane of focus (”focused particles”) are obviously different from
the particle images from planes that are not in focus, or ”defocused particles”1
. Focused particles
images are small and bright, i.e. they consists of 3-5 pixel objects and include saturated pixels of
the maximum image gray level (in our case, for 8 bit images: I = 28
− 1 = 255). In addition to
the saturated pixels, there are several neighboring pixels which belong to the same particle images,
however their brightness (gray level intensity) is significantly smaller. We found that an additional
threshold of the gray level intensity introduces too much noise, and we decided to identify the particle
images by using morphology image reconstruction (analogy to the region growing or propagation
algorithms): We define such objects as follows:
1. At the first stage (zero iteration) the saturated pixels are selected:
I(0)
= {x ∈ I |I(x) = 255}
2. Image reconstructing algorithm makes use of the identified objects as a ”marker” image and
1Note that there are two kinds of defocused particles, illuminated by two different laser sheets

0 2 4 6 8 10 12 14
0
50
100
150
200
250
300
Pixels
Intensity
Identified
Original
Reconstructed
Figure 5.10: Schematic view of the reconstruction principle used in the region growing algorithm:
(- -) Dashed line shows the one dimensional signal, (− · −) line is for the identified saturated pixels
and, (− • −) line presents the reconstructed object.
the enhanced PIV image as a ”mask” image to define the real boundaries of the particle image
Ifocus by iterative conditional dilation procedure. The n iteration image I(n)
is calculated as
follows:
I(n)
=

I(n−1)
⊕ dB

∩ I , I(n)
6= I(n−1)
while the iterations are repeated until there is no change between the images. dB denotes
the small structure element, such as circular element of 1 pixel radius, or 3 × 3 pixels square
element. Figure 5.10 schematically describes the reconstruction or region growing principle:
the identified binary image propagates toward the original image but does not pass the object
boundary.
3. The focused particle image (i.e. object) has to be small, therefore we can filter out the objects
with size larger than the threshold, TA. This area-based filtering procedure was performed
using gray level morphology opening, apparently the fastest and most efficient filter in this
case.

Focus particles filtered image
50 100 150 200 250
50
100
150
200
250
Figure 5.11: Image with particles in the focus plane.
The result of the method described above is presented in figure 5.11. The image shows the gray
level image with the objects on the flat, zero-level background. Subtraction of the focused image
from the multi-plane image is the next stage of the plane discrimination procedure. The objects that
are defined as focused particles are subtracted from the original image, and the removed pixels are
filled by the locally smoothed pixels, using ”top-hat” operator with circularly-shaped structuring
element of 3 pixels radius. An example of the ”defocus image” that contains two defocus planes is
shown in figure 5.12, together with the original image for comparison. Note that the defocus image
on the right does not include bright and small focused particles, but it is gray scale image without
sharp discontinuities.
5.6.3 Discrimination between two defocus planes
The discrimination between defocused particles, in two well-defined planes, is based on the object
property (size) based segmentation, i.e., separation between small and large objects. The implemen-
tation of the segmentation algorithm consists of two main steps: (i) definition and identification of

Enhanced XPIV image
50 100 150 200 250
50
100
150
200
250
Defocus particles image
50 100 150 200 250
50
100
150
200
250
Figure 5.12: Original (left) and defocus planes image (right).
the objects, and, (ii) classification (segmentation) of the objects into the two clusters (groups) based
on the size parameter. Originally, we expected that it would be possible to discriminate between de-
focused particles using additional parameters, such as intensity, gradient magnitude, etc. However,
the experimental components (imaging and laser optics, etc.) and setup reduced the significance
of these parameters. Thus, only the size or area parameter was found to be a good discriminate
characteristics of the specific particle images.
Object definition and identification
The particle image objects are defined and identified using a gradient-based segmentation procedure.
The gradient was calculated by using morphology gradient method and Canny’s gradient method
(Canny, 1986). Both methods provided robust and sharp results for the presented PIV images
(figure 5.13), and could be used interchangeably. In addition, gradient surfaces were handled as
gray level images. This kind of treatment facilitates to obtain the well defined, contrast objects on
the background, instead of object edges image, by filling the high-gradient disks and enhancing the
gradient maps. Enhancement and filling operations were implemented with gray level morphology
procedure. Figure 5.14 demonstrates the original and enhanced gradient images. Enhanced gradient
images were used to identify objects in the defocused planes by using following procedure:
• The gradient image was thresholded by using contrast thresholding procedure in order to select

Defocus particles image
50 100 150 200 250
50
100
150
200
250
Image gradient map
50 100 150 200 250
50
100
150
200
250
Figure 5.13: Defocus planes image (left) and gradient map as a gray level image (right).
Image gradient map
50 100 150 200 250
50
100
150
200
250
Enhanced gradient map
50 100 150 200 250
50
100
150
200
250
Figure 5.14: Gradient image (left) and enhanced gradient map (right)

only objects with the strong gradient;
• The ”broken edges” were connected by morphology closing with line structuring elements;
• Connected gradient borders were filled by morphology smoothing operator;
• The image was segmented based on the gray level intensity thresholding.
The result of this identification procedure is the binary image that includes 1’s at all locations
that are identified as objects (”true” Boolean values) and 0’s at all other locations (”false” Boolean
values). The resulted binary image (shown in figure 5.15), facilitates one to use the fast mathematical
morphology (binary) operations.
50 100 150 200 250
50
100
150
200
250
50 100 150 200 250
50
100
150
200
250
Figure 5.15: Defocus particles image (left) and the identified objects in a binary image (right).
Classification of objects
Using as an input the binary image, calculated by using gradient-based segmentation method, we
segment the identified objects into the two clusters, based on their size. The segmentation is based
on the fixed size threshold value, which is considered by using the granulometry technique. This
technique is implemented by means of the iterative morphological opening of the binary image with
an ascending set of the identical structuring elements, and by counting the number of removed pixels
at each iteration:
Sn
B =
X
I
{(I ◦ (n)B) − (I ◦ (n + 1)B)}

0 1 2 3 4 5 6 7 8 9 10
0
500
1000
1500
2000
2500
3000
3500
n
S
n
B
0 1 2 3 4 5 6 7 8 9 10
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
n
∂
S
n
B
/
∂
n
Figure 5.16: Size distribution (granulometry) of the binary image (left) and its derivative (right).
where nB = B ⊕ B ⊕ B . . . ⊕ B
| {z }
n
, and
P
I
produces an estimation of the area of the objects removed
at the n iteration. Note that the objects in the real image are not perfect geometric shapes, and
some of object boundary pixels are removed during iterations for the small n values. However, the
object of the specific size nB is removed completely at that iteration, and significantly changes the
Sn
B value. Figure 5.16 presents the size distribution of the defocused image, estimated by Sn
B, and
based on the circular structure element. In addition, the plot of the derivative ∂Sn
B/∂n indicates
that the image contains two separable populations of the objects with the size of 1B and 4B. The
defocused images were separated into the two planes by using two area-based filtering operations.
5.7 Combined PIV and HFIR experiment
The Hot-Foil Infra-Red Imaging (HFIR) measurement technique is described in the literature review
(see section 2.3.2). This non-intrusive measurement method makes use of an infrared imaging of a
heated foil located at the bound wall and explained the flow velocity at the most near-wall region as
they affect the temperature field of the foil surface. In the present work the combined measurement
system is introduced as it is shown schematically in the figure 5.17. In this system the temperature
field is measured from the bottom side of the flume by an infrared camera, and the PIV camera

Figure 5.17: Schematic view of the combined PIV - HFIR experimental setup.
.HFIR -e PIV z‘ zalynd dhiyd ly iznikq xe‘z :5.17 xei‘
acquired the particle images at the plane parallel to the flume from the upper side. Captured two-
dimensional images of the temperature field provide indirect visualization of the velocity field at the
wall, while the PIV provides the direct velocity field measurements. The disadvantage of the current
configuration is the relatively high distance between the bottom of the flume (i.e., the foil) and the
laser plane of the PIV system. This is mainly due to the inherent problem of the high reflection of
the metallic foil and it will be solved by replacement of the foil by IR transparent material with thin
conductive and antireflection coatings.
The infrared acquisition system consisted of an infrared scanner (Inframetrics 740), S-VHS video
recorder, computer, monitor and 8-bit frame grabber. The radiometer has a typical minimum
detectable temperature difference of 0.1◦
C, a time response of 25 fps and a horizontal resolution
of 256 physical pixels per line. The thin foil of 50µm, with negligible thermal inertia has very low
temperature difference between the two sides of the foil (about 0.1◦
C, as shown by Hetsroni and
Rozenblit (1994)).
Another technical limitation is that the utilized IR scanner lacks of the synchronization and

digital image transfer options. The infrared images, captured by the specific IR camera is recorded
analogically, and digitized by using the PC-based analog-to-digital converter, similar to the work of
(and with the help of first author) Kowalewski et al. (2000). As a result the velocity fields, as they
measured by the PIV system at the frame rate of 15 Hz, and the IR images, recorded at the 25 Hz
frame rate are not recorded exactly at the same instantaneous time points. However, the PIV and
IR images are synchronized at their first and last images by opening and closing the experiment at
the same time. Therefore, the post-analysis of the thermal and PIV images is performed separately
and the results presented in this work are only on the statistical basis, that is, by using the POD
approach to the velocity and the temperature fields.
5.7.1 Preprocessing of the thermal images
The recorded IR images have 256 intensity levels and have been recorded on video tape of the S-
VHS format as a conventional ”interlaced” video scan pattern of 25 frames per second. The video
was then used in a playback mode to and the video frames were captured and digitized by using
the DT-3155 frame grabber and stored as 768×576 pixels images with the 256 gray levels. The
example of the IR image recorded on the videotape is presented in its transformed, digital version
(i.e, the intensity map), in figure 5.18. The image includes two label rows with the parameters of
the experiment: the day, the camera type and the time at the upper row; and the temperature scale,
image mode and the average temperature at the bottom text row. In addition, below the bottom
text, the block of the gray levels from the darkest (zero level) to the brightest (255 level) is shown
for the specific picture and these levels are for the given in the text row temperature scale. In each
experimental setup, the temperature scale was kept constant, and its relative gray level scale was
used in the pre-processing stage. In addition, on the right side of the image we can see the bright
circles of the hot conduction connectors, and the flow was from the left to right, resulting in the
long and thin paths of the high-low intensity, translated to the low and high temperature streaks,
respectively.
First, the image borders are removed and the resulting image with the useful area representing
thermal image, which covers 740×450 image pixels and includes only the related foil surface area,
as shown in figure 5.19. One can notice the speckle (i.e., random noise) nature of the images, and
this due to the relatively low quantum efficiency of the IR camera and low infrared signal of the

Figure 5.18: Infrared image of the temperature field of the foil surface
.mnegnd ghynd ly dxehxtnhd dcy ly IR-d zpenz :5.18 xei‘
Figure 5.19: Trimmed image of the temperature field.
.dxehxtnhd dcy ly dkezg dpenz :5.19 xei‘

Figure 5.20: Enhanced image of the temperature field.
dxehxtnhd dcy ly zllkeyn dpenz :5.20 xei‘
foil: the temperature of the foil was kept at the level of 1◦
-2◦
above the temperature of the water,
in order to prevent the heating surface effect on the flow. Thus, the image is filtered by using the
common low-pass filter and after that enhanced by the same image processing algorithm, as one
described in the XPIV section 5.6. The filtered and enhanced image is shown in the figure 5.20. The
main goal of the enhancement is to stretch the histogram of the gray level image, and therefore, to
emphasize the difference between the relatively low and high temperature streaks, clearly presented
in the enhanced image. On the right side, the gray levels bar presents the 0-1 scale (i.e. 255 →
1) and allows to convert the intensity map into temperature map, shown in the figure 5.21 for the
subsequent analysis. This image depicts the temperature field of the foil surface with the appropriate
color levels, shown in ◦
C units.
The afterwards analysis of the combined PIV and IR technique is performed according to the
following procedure:
• PIV images are analyzed by InsightTM
software and velocity fields are transferred to Mat-
lab r
;
• IR images are saved as the temperature fields, according to the above image processing and
calibration techniques;

Figure 5.21: Temperature field image.
miizin‘d mikxrl leik mr dxehxtnh dcy :5.21 xei‘
• Coordinates (i.e., x1 and x3 locations), velocity components (i.e., ũ1 and ũ3), and temperature
fields T are scaled to the non-dimensional wall units x+
i , u+
i , T+
;
• Velocity derivatives
∂u+
1
∂x+
3
and temperature maps T+
are analyzed separately by the POD
method, described in the section 6.2. The results are compared qualitatively and quanti-
tatively afterwards and presented in the Chapter 6.
It is noteworthy that more quantitatively accurate analysis of the IR images is proposed by
using the Optical Flow Velocimetry (Kowalewski et al., 2000). The analysis of the temperature
fields recorded during the current work is under progress in the cooperation with Prof. Kowalewski
and the convective velocity fields of the temperature streaks and the directly measured velocity fields
will be compared in this analysis.

Chapter 6
Results and discussion
In this chapter a rather detailed description will be given of experimental results, as obtained by using
PIV in two and three component measurements (i.e., SPIV), PIV with HFIR combined technique,
and POD analysis. These results agree qualitatively and quantitatively with features described in the
literature review (see section 2.2.2) in a reasonable way. All results point towards a coherent features,
consisting of characteristics which appear to be repetitive (i.e., coherent) in nature, statistically
speaking, both in time and space. These coherent structures show streamwise oriented regions of
strong vorticity, formed an inclination angle with the streamwise direction in the x1 − x2 plane,
and with a consistent spanwise spacing (λ3) in the x1 − x3 plane. We show that associated with
these structures, the turbulence Reynolds shear stress (uiuj), vorticity (ωi), rate of strain (sij), the
production (−huiujiSij), and dissipation (sijsij) terms have all strong nature.
6.1 Conventional turbulent boundary layer flow analysis, x1−
x2 plane
6.1.1 Velocity fields and distributions
Particle Image Velocimetry provides the instantaneous two dimensional velocity field maps of two
(PIV) or three components (SPIV). An example of the instantaneous fluctuating velocity field in
x1 − x2 plane is shown in figure 6.1.
59

CHAPTER 6. RESULTS AND DISCUSSION 60
0 0.2 0.4 0.6 0.8 1 1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6.1: Example of the instantaneous fluctuating velocity field, u1,2.
.u1,2 zcpezd zexidnd dcy ly dnbec :6.1 xei‘
Figure 6.2 shows the example of the velocity distributions (profiles) of the ũ1 and the ũ2 compo-
nents (on the left and right sides, respectively). Few of the profiles are clearly related to the ejection
phase, when the low momentum fluid element leaves the near-wall region, and sweep phases, when
high momentum fluid enters the region closer to the wall. This picture is similar to profiles, achieved
by Grass (1971) in a water channel (h = 50 mm, average streamwise velocity U1 = 0.245 m/s)and
in full agreement with results of Runstadler et al. (1963), Corino and Brodkey (1969).
Figure 6.3 clearly presents the typical ensemble averaged streamwise velocity field in x1 − x2
plane of the turbulent boundary layer. The streamwise average U1 component of the velocity is by
order larger than its wall normal component U2, and the average streamwise velocity posses the
well known distribution shape with the distance from the wall (i.e., x2). In order to compare it to
the well known log law of the turbulent boundary layer, figure 6.4 is presented with the average
streamwise velocity distributions for the five experimental cases. The fitted line presents the log-law
u+
= A log(y+
) + B, with A = 2.5 and B = 5.
The correlation between the streamwise and wall normal fluctuating velocity components is pre-
sented by using the joint probability density function (joint PDF) in figure 6.5. This plot presents
very clear that on statistical basis, there is much more positions where the negative streamwise

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 6.2: Instantaneous profiles of streamwise ũ1 (left) and spanwise ũ2 velocity components
(right)
.ũ2 -e ũ1 ly irbx zexidn qexit :6.2 xei‘
velocity fluctuation is coincident with the positive wall normal velocity fluctuation, and vice versa:
positive u1 is coincident much more with negative u2, than with positive values. This result ab-
solutely agrees with the need of the turbulent momentum transfer in the boundary layer, as was
pointed out in section 3.1.3).
The one-point correlation tensor of two velocity components is Reynolds stress tensor, and
the spatial distribution of the strong negative correlation presented above is clearly shown by the
Reynolds stress component −hu1u2i in the x1 − x2 plane. This one of the most important features
of the turbulent boundary layer flow (see chapter 3). Figure 6.6 presents the Reynolds stress com-
ponent map with color coding and scale on the right side. The boundary layer region is significantly
different from the free stream flow (i.e., up to x2/h ≈ 0.4), and within the boundary layer region,
there is clear evidence of the region, responsible for the most intensive momentum transport - the
buffer zone.
Another important feature of the turbulent flow is its kinetic energy u2
i . Following figures 6.7
and 6.8 present the turbulent kinetic energy in streamwise and wall normal directions, respectively.
In these figures we should acknowledge the region of the intense kinetic energy in both directions
and this is the same region, responsible for the strong Reynolds stresses. The connection between
the Reynolds stresses and the kinetic energy is presented here by using the kinetic energy production
term, −2hu1u2iS12 (see section 3.2, shown in figure 6.9.

0.44 0.45 0.46 0.47 0.48 0.49 0.5
1
2
3
4
5
6
7
8
9
10
x 10
−3
x/h
y/h
〈 u 〉
Figure 6.3: Ensemble averaged velocity field. Note the streamwise velocity profile.
.zrvennd zexidnd dcy :6.3 xei‘
As it was expected, the region which is most responsible for the production of the turbulent
kinetic energy, which is, in order, responsible for the turbulence reproduction (self-sustaining), is
the same buffer zone at x2/h ≈ 0.1. In order to present this result in more quantitative way we
depict the result in the figure 6.10, similar to the analysis of the energy budget by Klewicki (1997).
This figure demonstrates that the kinetic energy production term is very strong around x2/h ≈ 0.1
and the main contribution to this term is the triple correlation between the u1, u2 and the derivative
of the average streamwise velocity in the wall normal direction, ∂U1
∂x2
.
It is obvious that the velocity derivatives play a dominant role in the turbulent flow. It is
transparent both from the theoretical basis (e.g., Chapter 3) and from the above analysis of the
turbulent flow PIV measurements. There are two interrelated quantities of the turbulent flow,
instituted by the velocity derivatives, namely vorticity ωi and rate-of-strain, sij. In the following
figure 6.11 we present the results achieved in the x1 −x2 plane. These two quantities in this plane are
almost1
identical due to very strong streamwise velocity derivative in the wall normal direction ∂U1
∂x2
,
much more substantial than the ∂U2
∂x1
. The presence of the strong shear near the wall is important in
1Previous figure 6.10 presents that there is some small difference between them.

Figure 6.4: Streamwise velocity distributions in wall units, along with the log-law line.
inihxbeld wegd ly ewe xiw zecigia rvenn zexidn qexit :6.4 xei‘
u’
1
u’
2
Joint PDF
Figure 6.5: Joint PDF between u1 and u2
u2 dna u1 zexidnd iaikx ly joint PDF zdivwpet :6.5 xei‘

−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
x 10
−4
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/h
y/h
〈 u
1
u
2
〉
Figure 6.6: Reynolds stress hu1u2i
hu1u2i Reynolds ivn‘n dcy :6.6 xei‘
1
2
3
4
5
x 10
−4
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/h
y/h
〈 u
1
2
〉
Figure 6.7: Streamwise kinetic energy u2
1.
dnixfd oeeka zihpiw dibxp‘ :6.7 xei‘

2
3
4
5
6
7
8
9
10
11
12
x 10
−5
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/h
y/h
〈 u
2
2
〉
Figure 6.8: Wall normal kinetic energy u2
2.
xiwl avip oeeka zihpiw dibxp‘ :6.8 xei‘
0
1
2
3
4
5
6
7
8
x 10
−4
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/h
y/h
−2〈 u
1
u
2
〉 S
12
Figure 6.9: Turbulent kinetic energy production −2hu1u2iS12
−2hu1u2iS12 zihpiw dibxp‘ zxivi :6.9 xei‘

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−7
−6
−5
−4
−3
−2
−1
0
x 10
−4
y/h
−〈
u
v
〉
dU/dy,
−〈
u
v
〉
S
12
−〈 u v 〉 dU/dy
−〈 u v 〉 S12
Figure 6.10: Turbulent kinetic energy production −2hu1u2iS12 versus wall normal coordinate.
.xiwl zavipd dhpicxe‘ew cbpk −2hu1u2iS12 zihpiw dibxp‘ zxivi :6.10 xei‘
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1 1.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6.11: Ensemble averaged vorticity ω3 (left) and strain S12 (right).
mixeairde zeileaxrd ly mirvennd zecyd :6.11 xei‘

−10 −5 0 5 10 15 20
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.002
0.007
0.012
0.017
0.022
x [m]
y
[m]
ωz
Figure 6.12: Instantaneous vorticity ω3 component field.
.zirbxd zeileaxrd dcy :6.12 xei‘
the connection to the turbulent energy production term −hu1u2iS12. Actually, this term represents
the rate at which the energy is transferred from mean flow to turbulent fluctuations. We already
noted that this term appears in the mean flow energy equation (3.16) and proposes that the turbulent
energy is generated from the mean shear. In addition, we present in the following analysis that the
strong mean vorticity (or strain) performs a masking effect on the concealed coherent structure of
the boundary layer. When we look at the instantaneous vorticity fields near the bottom wall, as one
presented in the figure 6.12, we can detect a large region of the concentrated vorticity, elongated in
the streamwise direction and inclined at some small angle (i.e., between 5◦
and 15◦
). Such structures
appear randomly in measured fields and at different streamwise locations, relatively to the measured
flow field boundaries.
6.2 Linear combination of the POD modes
The largest eddies do have directional preferences, and their shapes are characteristic of
the particular mean flow. The recognizable eddies are called ’coherent structures’. The
allusion is to human ability to recognize the forms, rather than to statistical concepts of
coherence. Durbin and Pettersson Reif (2001)

6.2.1 POD of velocity/vorticity
Section 2.4.4 lists the publications about the POD analysis that almost always has been applied to the
fluctuating velocity fields ui in certain types of flows, such as jets, boundary layers, backward facing
step flows, etc. The basic assumption of the velocity analysis is that the large coherent structures
in these flows contain the main fraction of the turbulent kinetic energy, u2
i . As it is pointed out
by Kostas et al. (2001) and Liberzon et al. (2001), the velocity field in turbulent boundary layer
flow is less suitable for the POD-based coherent structure identification technique, due to the strong
dependence on the mean velocity (i.e, jitter effect). Instead, we propose the POD analysis of the
scalar fields of the vorticity vector components measured with PIV (including SPIV and XPIV ), as
an important Lagrangian invariant quantity of the turbulent flow.
6.2.2 Comparison of the POD of the velocity and vorticity
As was shown by Kostas et al. (2001), the curl operator on the modes of the fluctuating velocity
field not necessarily reproduces the relative modes of the fluctuating vorticity field. However, the
coherent motions that repeat themselves with some regular spatial characteristics, should have their
footprints observable both in the velocity and in the vorticity fields. Figure 6.13 shows that there
is clear evidence of the large scale coherent motion, presented by the first mode of the velocity and
vorticity fluctuating fields. The parallelism is shown by gray scale coding of the curl of the velocity
and vorticity modes.
The same kind of similarity is shown by second modes (see figure 6.14) and was observed in
following modes. The similarity is not always presented with such strong parallelism as in above
figures, but linear combination of POD modes of both quantities have shown very similar spatial
structures of coherent patterns. The vorticity is found to be more suitable for the POD analysis, due
to the stronger relevance to the coherent structure identification method: ”a large-scale turbulent
fluid mass with spatially phase-correlated vorticity”, Hussain (1986), Chapter 2.
6.2.3 Average field and the first POD mode
Karhunen-Loéve decomposition theorem was developed for the zero-mean random fields and this is
the way it was introduced to the turbulence research community by Lumley (1970). Generally, there

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25
0
0.125
0.25
0.375
0
0.125
0.25
−0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25
0.0625
0.125
0.1875
0.25
0.3125
0.0625
Figure 6.13: First POD mode of the fluctuating velocity field (left) and vorticity (right)
.)oini( zeileaxrd lye )l‘ny( zexidnd ly oey‘xd cend :6.13 xei‘

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25
0
0.0625
0.125
0.1875
0.25
0.3125
0.375
−0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
0.0625
0.125
0.1875
0.25
0.3125
Figure 6.14: Second POD mode of the fluctuating velocity field (top) and vorticity (bottom)
)oini) zeileaxrd cene )l‘ny) zexidnd ly ipyd -d cen :6.14 xei‘

should be no difference between the POD modes of the fluctuating velocity (or vorticity) fields (i.e.,
after the average velocity is subtracted from the instantaneous fields) and the instantaneous velocity
fields, except their numbers come with the phase. It is clear that for a random and stationary signal,
the first mode should be its statistical average. In addition, the following modes should reduplicate
the modes, if the average was calculated and subtracted before the decomposition analysis. The
validated PIV measurements (Chapter B) present that the mean flow is statistically averaged and,
therefore, being time-independent, it is orthogonal to the POD modes.
However, we found that for finite number of PIV realizations, this is not necessarily the case,
similarly to the observations of Aubry et al. (1988). In most cases, the first POD mode of the
instantaneous vorticity (or velocity) data represented the field, which is just proportional to the
mean vorticity field, as it is shown in figure 6.15. Moreover, we found that POD method is an
accurate outlier detection filter. In the case when we impose one erroneous velocity map in the
set of several hundred PIV realizations, the first POD mode included the absolutely random mode
with very high energy fraction, instead of the expected average velocity field. The related random
coefficient a(k)
is significantly higher for that specific outlier velocity data. Second, and following
modes, include the average velocity field and the same POD eigenfunctions as without the erroneous
field, respectively.
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1
Figure 6.15: POD modes of the instantaneous (ω̃3, left) and fluctuating (ω3, right) vorticity fields.

6.2.4 Symmetry of the orthogonal decomposition
It has been recognized that if the two point correlation kernel Rij(x, x0
), used in POD transform,
is homogeneous in some direction, or invariant under translation (i.e., Rij(x, x0
) = Rij(x − x0
)), it
has a simple Fourier series representation. This means that the two-dimensional eigenvalue problem
can be reduced to a one-dimensional problem (Aubry et al., 1988, Berkooz et al., 1993), and the
correlation kernel can be replaced by its Fourier transform in the homogeneous directions. However,
in a turbulent boundary layer, this decomposition produces non-localized spatial structures in the
homogeneous direction, unlike the observed instantaneous flow features. Moin and Moser (1989)
presented the pair of streamwise rolls in a turbulent channel flow, as leading empirical eigenfunctions,
from the DNS results by Kim et al. (1987). The rolls have some spatial characteristics, such as
diameters of few tens in wall(+ system) units, appeared approximately in the streamwise direction,
occur alone or within a counter-rotating pair. Holmes et al. (1996) proposes that the homogeneity
of the flow generated symmetric kernel, and constrained the rolls to appear symmetrically with an
equal strength. Rempfer and Fasel (1994) observed empirical eigenvalues and eigenfunctions that
occur in pairs, and related this observation to the approximate symmetry of the two-point correlation
kernel, due to the slowly growing structures in the transition region of a flat plate boundary layer.
We present in our results (Chapter 6) similar observations of symmetric appearance of spatial
eigenmodes in the PIV data. The two-point correlation is only approximately symmetric, accordingly
to the observation that homogeneity might occur in spatially unbounded systems or systems with
periodic boundary conditions (e.g., Holmes et al., 1996). In the case of the PIV experiments (i.e.,
the unbounded system without periodic boundary conditions) we cannot develop our correlation
kernel in a Fourier series, but we might take into account the symmetry option in the analysis of the
eigenmodes. Figure 6.16 presents the plot of the relative energy fraction of the first ten eigenmodes
of the fluctuating vorticity field. The relative fractions of the energy of first 8 modes, which is defined
by equation 2.23, are 1.96%, 1.81%, 1.68%, 1.56%, 1.48%,1.45%,1.37%, and 1.35%, respectively. It
is clear from the analysis of the eigenvalues, that modes 5 and 6, 7 and 8, and others are more likely
to be pairs, rather than single modes. Such similarity of the eigenvalue magnitudes is expected, if
we adopt the assumption of homogenuity (Holmes et al., 1996, Section 3.3.3):
Rij(x − x0
) =
N
X
i=1
cke2πik(x−x0
) (6.1)

0 50 100 150
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
k, mode index
λ
k
/
Σ
λ
i
2 2.5 3 3.5 4 4.5
0.046
0.047
0.048
0.049
0.05
Figure 6.16: Symmetric modes evidence in ”energy” spectrum of the decomposition.
ilpbehxe‘d wexitd ly zenverd qexita miixhniq micen ly iedif :6.16 xei‘
and recognize that each single kth
eigenfunction is represented by sum of the sin and cos functions
with equal coefficients of half of the ck coefficient. In order to check our prediction about the
similarity of identified pairs of modes, we present them separately and in their linear combinations.
It is worth to note that we have not applied any weighting to the eigenmodes, and summarized them
directly with the aim to analyze their spatial characteristics only. We notice that the magnitudes
of the eigenvalues 5 and 6 (and 7, 8) are very similar, and their spatial properties explain this
phenomenon. The linear combination of modes 5 and 6 is shown in figure 6.17 along with the
separate modes 4, 5 and 6. One can see that the spatial form of their sum is almost exactly
represent the mode 4, thus does not introduce any new information about the spatial properties
of the large scale coherent features. Similar conclusion we derived from the analysis of the linear
combination of modes 7 and 8.

a) −0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.005
0.01
0.015
0.02
0.025
0.03
b) −0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.005
0.01
0.015
0.02
0.025
0.03
c) −0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.005
0.01
0.015
0.02
0.025
0.03
d) −0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.005
0.01
0.015
0.02
0.025
0.03
Figure 6.17: Four (a), fifth (b), six (c), and linear combination of 5th and 6th (d) POD modes of
the fluctuating vorticity
:6.17 xei‘
6.2.5 Choice of eigenmodes
It was reported by Kostas et al. (2001) and Liberzon et al. (2001) that the ”energy fraction”
N
P
k=1
E(k)

contained in first POD modes (i.e., N = 3, 5, 10) of the fluctuating vorticity field
is not significant, as of the velocity POD modes. Therefore, we cannot truncate the insignificant
POD modes of the vorticity by selecting only the modes that contain 85% - 90% of the ’energy’ of
the data set (i.e, enstrophy hω2
i). In the presented work, we select the vorticity modes only by their
contribution to the spatial form of the coherent structure. We have shown that there are modes
that are just symmetric counterparts of the others, and that these modes do not change the shape
of the identified structure.
We can refer this kind of treatment as one similar to the pattern recognition analysis, when the
iterative search stops when the template does not change its shape. Figure 6.18 depicts the linear
combination of the first 3, 5, 10 and 150 modes, from the top to the bottom (according to the
equation 4.1). The spatial localized coherent structure is clearly shown to be streamwise elongated

region of the condensed vorticity that does not change its spatial characteristics by addition of 5, or
10, or 150 modes. Comparison of the representation in three modes with that shown by 5, 10, and
150 modes reveals that the overall representation of the structure by only the first three vorticity
POD modes is reasonable to define the spatial characteristics, such as streamwise length, angle and
wall normal location of the structure.
The most important advantage of the POD technique presented here is the unbiased and objective
approach to the extraction of the underlying features.
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x 10
4
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.005
0.01
0.015
0.02
0.025
0.03
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
0.005
0.01
0.015
0.02
0.025
a
b
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x 10
4
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
0.005
0.01
0.015
0.02
0.025
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0.005
0.01
0.02
c
d
Figure 6.18: Linear combination of the POD modes of the fluctuating vorticity ωz component, a)
3, b) 5, c) 10, and d) 150 modes, respectively.

6.2.6 Vorticity component ω3
According to the analysis approach (Chapter 4), the statistical analysis of the data ensemble, that
keeps its spatial characteristics and not affected by the basis functions (e.g., sinusoidal functions in
Fourier transform) should be performed by using the Proper Orthogonal Decomposition (POD). In
addition, we presented that the underlying coherent structure is appropriately extract by employing
the linear combination of the POD modes of vorticity. Figure 6.19 depicts the linear combination of
three first POD modes of the vorticity component fluctuations ω3 as it is measured and calculated
in the x1 − x2 plane. The replication of all spatial characteristics of the coherent feature is quite
remarkable: the length, the inclination angle and the spatial location of the extracted feature are
the same for only the combination of the first thee POD modes: two large-scale coherent patterns
with different sign of the vorticity values appear side-by-side in the near wall region (10 ≤ y+
≤ 100)
of the flume.
−0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.125 0.375 0.625 0.875 1.125
0.0625
0.125
0.1875
0.25
0.3125
0.375
Figure 6.19: Linear combination of three POD modes of the vorticity component, ω3.
.ω3 zeileaxrd aikx ly -d icen 3 ly zix‘ipil divpianew :6.19 xei‘
In the previous section 6.1 we presented the similarity between the average vorticity field and the
ensemble average of the rate of strain tensor component S12 due to significantly large derivative ∂U1
∂x2
.
The similarity is not obvious in the instantaneous fields, but it is authoritatively presented in the
linear combination of POD modes of the fluctuating fields of the strain, as it is shown in figure 6.20.

−0.06 −0.04 −0.02 0 0.02 0.04 0.06
0.25 0.5 0.75 1 1.2 0.25 0.5 0.75 1 1.2
0.005
0.01
0.015
0.02
0.025
0.03
Figure 6.20: Linear combination of three POD modes of the rate of strain component S12.
.S12 zxeaird avw ly -d icen 3 ly zix‘ipil divpianew :6.20 xei‘
6.2.7 Parametrization of the results
In order to analyze the influence of the flow parameters on the extracted coherent structure of the
turbulent boundary layer, experiments were carried with several different Reynolds (Reh) numbers.
Following figure 6.21 presents the linear combination of the vorticity POD modes for experimental
results of four different cases: Reh = 24000, 27000, 45000 and 540000. The constant parameter was
the height of the flow, and we varied the flow rate of the water at the inlet.
Analysis of these results (figure 6.21) shows the consistency of the spatial characteristics of the
identified features: the streamwise non-dimensional length (x+
≈ 600 − 1000) and the average
expansion angle (8.5◦
). This characteristics shows an impressive correspondence to the coherent
features recently disclosed by Jeong et al. (1997) in a numerically simulated turbulent channel flow
and in the experimental results of Kaftori et al. (1994).
The consistent topology characteristics of the spanwise vorticity component ω3 fields are ex-
plained as the footprints of a coherent structure elongated in the streamwise direction. It is note-
worthy that the identified features almost in all cases include a pair of two concentrated regions of
the opposite-sign vorticity component, that is +ω3 and −ω3, presented in figure 6.21 as very bright
and very dark color neighbor regions. One should keep in mind that these are two dimensional

images are of the vorticity component ω3, or more accurately, the projection of the vorticity vector
~
ω on the x1 − x2 plane. Thus, by utilizing the simple geometrical description, this is due to repro-
ducible, regular structure with large scale areas of the relatively strong vorticity, not parallel to the
Cartesian coordinate axis, x1, x2, x3. This way the rotational motion regions can be explained by the
presence of a quasi-streamwise structure, which is elongated in the streamwise direction and inclined
upward from the wall. This coherent feature grows from the solid boundary and includes high-level
vorticity spots during its development, thence provides a full description of all known features, such
as streaks, ejections and bursts.
6.2.8 Identified coherent structure - discussion
As it was pointed out in the theoretical background section ( Chapter 3) and as it is described in
the textbooks of Tennekes and Lumley (1972) and Durbin and Pettersson Reif (2001), the simplest
analysis of the boundary layer flow in x1 − x2 plane shows that in the parallel shear layer the rate
of strain is: 


0 1
2
∂U1
∂x2
1
2
∂U1
∂x2
0


 (6.2)
Under this simplification of the problem, the rate of strain tensor has two eigenvectors, proportional
to (1,1) and (-1,1). This is the origin of the analysis that shows that these eigenvectors are at ±
45◦
to the wall, and this is the fact that is used as a prove of the ∼45◦
of the attached eddies of
Townsend (1956) and after that by many researchers (see for example Zhang et al., 1997). This
analysis suggests that these large eddies tends to align with the principal directions of strain, and
efficiently extract energy from the mean flow. At this point we should look again at the figure 6.20.
This is the real, accurately measured direction of the principal axis (i.e., eigenmodes)
of the strain in the turbulent boundary layer. We can explain the difference between the
theoretical angle (from simplified two dimensional analysis) and the smaller angle in our results by
the fact, that the average values of the derivatives ∂Ui
∂xj
are lower than the dominant ∂U1
∂x2
, but are
definitely not zero. We believe that this is the actual reason of the angle smaller than 45◦
between the
principal axis of the strain and the wall. In addition, we may call attention to the required balance
of forces: a mean shear rotates a material lines toward the streamwise direction (i.e, to make them
parallel to the wall), and at the same time, the vortical structures has self-induced velocity that

lifts it away from the wall, clearly shown by nonzero ∂U2
∂x1
. To achieve some inclined orientation, the
vortical structure has to live in some balance between rotation by the mean shear and lifting velocity.
This fact should explain that only coherent structures with sufficiently large vorticity, similar to the
identified in this work, can be self-sustaining.
6.2.9 Results from the x1 − x3 plane
In the previous section, the projection of the coherent structure on the x1 −x2 plane has been shown,
and due to the limitation of the two dimensional PIV and SPIV systems, the spanwise direction was
not considered then. Clearly, that the structure should present some coherent spanwise behavior, and
the interference with neighbor structures. Thus, if one accepts the result of the vortical structures,
lifted away from the wall and elongated in the streamwise direction, than it is obvious that the fluid
close to the wall surface between two neighbor structures should be compressed and ejected from the
wall. Between vortical structures, low speed streaks are observed near the surface, as we present in
the figure 6.22. The long regions of the negative streamwise velocity fluctuations (u1 0) are clearly
seen in the contours and the overall picture of the velocity fluctuations is presented by vectors. The
flow between the eddies should move upward from the surface, and it convects the low momentum
fluid away from the surface and recognized as an ejection. We can refer at this point to the analysis
in the book of Nezu and Nakagawa (1993) and references to the experimental results therein (see
p. 93) that proposes that the strong up- and downward flows (i.e., strong u2) is presented at the
regions of negative and positive regions of the ∂(u1u3)
∂x3
, respectively. Figure 6.23 presents the result
that clearly shows the effect, depicted by Smith (1984): the ejection of the low momentum fluid
upward from the wall.
This figure presents the gray level contour patches of the Reynolds stress component gradient ∂u1u3
∂x3
and two levels of the positive and negative streamwise velocity, that represents the high and low
speed streaks, respectively. We should notice the concurrence of the high gradient of the Reynolds
stress on the interface between the high and low speed streaks.
The importance of the measurements in the streamwise - spanwise plane (i.e., x1 − x3) is utmost
for the analysis of the spatial behavior of the coherent structures and their relation to the low
speed streaks (e.g., figures 6.22 and 6.23). In order to investigate the development of the spanwise
spatial characteristics as the structure lifts up from the wall, the measurements were performed

in the x1 − x3 configuration (see Chapter 5) at three different x2/h locations: 0.0375, 0.125, and
0.3125. The constant parameter was Reh = 27000. The analysis is according to our approach, that
is without thresholding or filtering, and it utilizes the linear combination of the POD modes of the
wall normal vorticity component, i.e.,
3
P
i=1
φ(i)
(ω2). All the results are presented on the same plot
with the goal to visualize their spatial properties (figure 6.24).
Figure 6.24 shows the dark and bright patches of the strong positive and negative vorticity
regions of the POD modes
3
P
i=1
φ(i)
(ω2) , respectively. The spatial order of the structures is very
clear, conjointly with the obvious enlargement of the structure scale at larger values of the x+
2 (i.e., far
from the wall). The remarkable attribute of the identified structure in this plane of interest is that the
spatial characteristics are in excellent agreement with the features found in x1 −x2 plane. Therefore,
we can assume, that the identified footprints are the projections of the three dimensional structure,
specified by strong vorticity vector projections. This assumption presents a notable correspondence
to the schematic view of the vortical structures presented by many researchers (Klewicki, 1997,
Smith and Walker, 1995), that grow from the pair of low-speed streaks, cause an ejection of the
low-speed fluid upward toward the mean flow, and include the concentrated region of the strong
ω2 between the streaks or the ”legs” of the structure. The development of such structure upward
from the wall is necessarily implies the change of scales. As we can notice from the figure 6.24, the
number of the opposite-sign vorticity strips reduces at the larger distances from the bottom. The
lowest plane, at x+
2 ≈ 30 contains four strips over the 400 wall units (see the axis tick labels), hence
the average spacing is measured to be approximately 100 wall units. This characteristics recites the
well known spanwise streak spacing λ+
≈ 100 since visualization studies of Kline et al. (1967).
6.2.10 Results from the x2 − x3 plane
The PIV measurements in the wall normal – spanwise (x2 − x3) plane of the flow is the most
complicated measurements setup (see figure 5.5), since the out-of-plane component of the velocity
is the streamwise velocity component, which is by an order of magnitude larger than the in-plane
velocity components. Therefore, the PIV measurements do inherently include more noisy data,
resulting in lower ratio of successful PIV realizations. Here we present the results acquired under
the conditions of case 8 in Table 5.1, and analyzed by using the
3
P
i=1
φ(i)
(ω1). POD of the streamwise
component ω1 is shown in figure 6.25. The topology of the symmetric, circularly shaped vorticity

regions proposes that vorticity projection on the x2 − x3 plane is originated from the strong vortical
motion with dominated streamwise direction (i.e., streamwise vortical structure). This coherent
feature apparently reproduces the streamwise vortical structures shown by Jeong et al. (1997),
Schoppa and Hussain (2000), among others. Again, it is noteworthy that the shape of the two
dimensional footprints in the x2 − x3 plane is agreeable with the form of spatial footprints as we
depicted above in two other orthogonal planes. The coherent structure appears to be the elongated,
inclined and tilted quasi-streamwise vortical structure.
6.2.11 Characterization of the structure by using the three-dimensional
reconstruction
In this section, the three dimensional picture of the assumed coherent structure is presented by
utilizing the reconstruction approach and the identified structure footprints from the orthogonal
plane measurements. Figure 6.26 is the combined plot of the results demonstrated in the figures 6.19,
6.24, and 6.25, when each one of the two dimensional contour plots is located on the appropriate
projection plane (i.e., planes denoted in descriptive geometry literature as π1, π2, and π3 planes,
or namely, ’top’, ’front’ and ’left’ views). We should avoid the misleading and point out that
the projection used for the top view is not actually measured at the bottom wall of the flume,
but at the buffer region, x+
2 ≈ 80. The inspection of the two dimensional footprints makes the
three dimensional view of the structure imaginable by using the reconstruction procedure. There
is a remarkable similarity between the spatial characteristics of all three planes. The structure is
streamwise directed, oriented at some angle upward from the bottom wall and has two sides (that
might be ’legs’ or two sides of one streamwise vortical structure) with opposite signs of the vorticity.
These observations are coinciding with the projection on the x2 −x3 plane, where the ’cross-section’
of the vortical structure is shown.
6.3 XPIV results and discussion
The XPIV extension of the stereoscopic PIV system allows to measure instantaneously the velocity
in three parallel planes. The XPIV results are validated first versus the previous PIV data by means
of ensemble averaged velocity profiles. The figure 6.27 presents the streamwise velocity distribution

(U1(x2)) by color lines, as it measured in several streamwise and spanwise locations by using the
XPIV . On the same plot, we present the results of the two dimensional PIV measurements of the
flow under same conditions, by means of ’boxplot’ graph. This plot shows that XPIV measurements
are absolutely overlaps the results measured in two dimensional PIV, and on the average, it displays
the same level of variance of the velocity distribution.
The results of the relative turbulence intensity (i.e., root-mean-square of the fluctuating velocity,
normalized by an average streamwise velocity component), is presented in figure 6.28 for the stream-
wise and spanwise velocity components, and for the three planes. The results are demonstrated by
imposing the XPIV results (black square) and the results of the two dimensional PIV measurements
(see Section 6.1), presented by points and interpolated spline. These results are in full agreement
with a known effects of the strengthening of velocity fluctuations close to the wall and the presented
figure is in good agreement with the classical data in turbulent boundary layer (Hinze, 1975), and
previous measurements in the flume (Kaftori et al., 1994, 1998).
In order to validate the measurement technique we applied the continuity equation for our data.
For an incompressible flow, the continuity equation can be written as:
∂u
∂x
+
∂v
∂y
+
∂w
∂z
= 0 (6.3)
We calculated the ∆U1/∆x1, ∆U2/∆x2, ∆U3/∆x3 (capital letters denote the ensemble averaged
velocity fields) and analyzed them to compare with the experimental results of Zhang et al. (1997, ,
Fig. 9a)), obtained by Holographic PIV. Zhang et al. (1997) proposed to use a normalized parameter
η as a quantitative measure of the quality of the velocity data measured with PIV technique:
η = 1 +
2 (∂U1/∂x1 ∂U2/∂x2 + ∂U1/∂x1 ∂U3/∂x3 + ∂U2/∂x2 ∂U3/∂x3)
(∂U1/∂x1)2 + (∂U2/∂x2)2 + (∂U3/∂x3)2
The distribution of the estimated values of η̄ is presented in figure 6.29. The resulted plot is
similar to the calculations of Zhang et al. (1997, see Fig. 10) and proposes that the XPIV data
approaches the quality level of Holographic PIV measurements, as it contains a sufficient number of
velocity realizations.

6.4 Coherent structures in XPIV results
6.5 Characterization of DNS data
6.6 Combined velocity/temperature footprints

200 400 600 800 1000 1200
20
70
120
170
220
270
200 400 600 800 1000 1200 1400
20
70
120
170
220
270
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
500 1000 1500 2000 2500
20
70
120
170
220
270
320
370
420
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
20
70
120
170
220
270
320
370
Figure 6.21: Linear combination of three orthogonal modes of the vorticity ω3 for a) Reh = 24000,
b) Reh4 = 27000. Linear combination of three orthogonal modes of the vorticity ω3 for c)
Reh = 45000, d) Reh4 = 54000.
ly micen .00072 = Reh4 )b ,Reh = 24000 )a :ieqipd i‘pz xear zeileaxrd aikx ly ly mice :6.21 xei‘
.00045 = Reh4 )d ,Reh = 45000 )c )a :ieqipd i‘pz xear zeileaxrd aikx ly

−0.06
−0.04
−0.02
0
0.02
0.04
0.06
−200 −150 −100 −50 0 50 100 150 200
0
100
200
300
400
500
600
700
800
900
Figure 6.22: Contour map of the streamwise velocity fluctuations u1 along with the vector plot of
the velocity fluctuations.
.zexidnd ly mixehwe htn mr cgi u1 zexidnd zeiv‘ehwelt ly mixehpew ztn :6.22 xei‘

−3
−2
−1
0
1
2
3
4
5
x 10
−4
−200 −150 −100 −50 0 50 100 150 200 250
0
100
200
300
400
500
600
700
800
900
Figure 6.23: Instantaneous field of the streamwise velocity fluctuations (red and blue line contours)
over the field of the ∂(u1u3)
∂x3
.
.∂(u1u3)
∂x3
zxfbpd ly dcyd rwx lr irbxd cpezd dnixfd dcy :6.23 xei‘

Figure 6.24: Linear combination of three POD modes of the wall normal vorticity component ω2
for x2/h = 0.3125 (top plane), 0.125 (middle), and 0.0375 (bottom plane).
.mipey mixeyin dyely xear xiwl avipd zeileaxrd aikx ly -d micen :6.24 xei‘
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
20
40
60
80
100
120
140
160
30
80
130
180
230
280
Figure 6.25: Linear combination of the POD modes of the streamwise vorticity ω1 component in
the x2 − x3 plane.
x2 − x3 xeyina ω1 zeileaxrd aikx ly -d micen :6.25 xei‘

Figure 6.26: Schematic view of the POD modes combinations as the projections on three
orthogonal planes. Note that x − z plane is at y+
= 100, the y+
axis is for the x − y and y − z
planes only.
miilpebehxe‘ mixeyin dyelya zelhdk micend ly ihnikq han :6.26 xei‘

0.2 0.21 0.22 0.23 0.24 0.25 0.26
0.1
0.5
0.9
Figure 6.27: Streamwise velocity average profiles measured by using XPIV (-o) and box-plot of the
PIV measurements in separate y planes(|-[]-|).
.PIV -e XPIV ir cecnd ,dnixfd oeeika zrvennd zexidnd qexit :6.27 xei‘

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Figure 6.28: Relative turbulent intensities u1/U1 and u3/U1 for planes from the XPIV and 2D PIV
measurements.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.05
0.1
0.15
0.2
0.25
0.3
η
f(η)
Figure 6.29: Distribution of η̄.

Chapter 7
Summary and Conclusions
7.1 Spatial characteristics
7.2 Further research
Durbin and Pettersson Reif (2001): Control of turbulence can be based on enhancing and destroying
the recognizable structure: breaking up long lived, large eddies can reduce drag by shorting the range
over which momentum is stirred.
However, the most intense vorticity in turbulent flows is associated with the smallest scales of
motion. This small scale vorticity is thought to be nearly isotropic in its structure. Such is not the
case of the largest eddies. Coherence, if it exists, is to be found in the large scales.
The largest eddies do have directional preferences, and their shapes are characteristic of the
particular mean flow. The recognizable eddies are called ’coherent structures’. The allusion is to
human ability to recognize the forms, rather than to statistical concepts of coherence.
The present experimental study of the topology of coherent structures in a turbulent boundary
layer is based on measurements of the three-component, two dimensional velocity fields, by using a
SPIV technique. The measurements were performed in a flume in three orthogonal planes, various
Reynolds numbers, and at several different locations. The unbiased statistical description of the
two dimensional footprints of the coherent motions, was obtained by means of POD of the vorticity
components. The linear combination of the first three POD modes of the spanwise component
91

CHAPTER 7. SUMMARY AND CONCLUSIONS 92
ωz, show the quasi-streamwise vortical structure of several hundred wall units length and inclined
upward with the angle of about 10◦
. The streamwise – spanwise cross section of the vorticity
fields present the spanwise spacing between the structures of ≈ 100 wall units and there is clear
evidence of the streamwise rotational motion in the y − z projection of the vorticity. The imaginary
combination of the orthogonal plane footprints in the three dimensional view proposes the existence
of the large scale, quasi-streamwise structure, elongated in the streamwise direction. The structure
is originated in the near wall region, characterized by a strong turbulent kinetic energy production
(−huviS12), and develops towards the outer region of the boundary layer. The identified coherent
features of the turbulent boundary layer have to be further investigated by using the proposed
statistical characterization method of the DNS data, or of the instantaneous three dimensional
velocity measurements.
7.3 Concluding remarks
New multi-plane stereoscopic PIV technique, XPIV , is proposed here. The technique is capable
of measuring statistically significant, three dimensional velocity data with the usual PIV spatial
resolution in two dimensions, and moderate resolution in the third dimension. The method makes
use of the basic principles such as simultaneous multi-plane illumination and the defocusing of the
scattered particle images. The three plane images were recorded simultaneously on two CCD cameras
and an advanced image processing techniques were applied to separate them into three pairs of PIV
images for each plane of illumination.
Qualitative and quantitaive measure of the velocity and velocity gradients data is demonstrated
by velocity profiles, turbulent intensity and the continuity equation. The experimental results and
analysis proposes that the XPIV measurement system is a suitable tool for the three dimensional
velocity measurements. The ability to achieve a three dimensional measurements will provide new
insight on turbulence research in general and on coherent structures in particular. The ability to
estimate experimentally the Navier-Stokes equations in its different forms (RANS, enstrophy, kinetic
energy, etc.) on one hand side and to characterize the flow patterns (vortices, bursting, etc.) on the
other side will enable one to determine the role of coherent structures in turbulent flows.
Further effort will be focused on refining the image processing algorithm, and improving the

CHAPTER 7. SUMMARY AND CONCLUSIONS 93
robustness of the technique.
Acknowledgements
The authors wish to thanks Dr. Miriam Zacksenhouse for her helpful review and comments on an
earlier draft.

Appendix A
Error estimation of PIV
experimental data
A.1 Mean velocity confidence level
The PIV data in this work is used only in the statistical way, it means that each flow property is
calculated on the base of certain number of PIV realizations, or samples:
Ui = hũii =
1
N
N
X
1
ũi (A.1)
where both ũi and Ui are matrices over the grid points x1, x2, x3 in a discrete version.
Sample size is the required number of realizations, N in order to achieve a required statistical
meaning of the statistical properties, such as an average in the equation A.1. From elementary
statistical analysis, confidence in the data is a function of the number of data samples N of the flow
field. The confidence limits for the mean (Snedecor and Cochran, 1989) is defined as:
U ±
tα/2,N−1 u0
1
√
N
(A.2)
where u0
is the rms of the streamwise velocity component, and tα/2,N−1 is the upper critical value
of the t−distribution with N − 1 degrees of freedom. The confidence coefficient is defined as 1 − α.
From the formula, it is clear that the width of the interval is controlled by two factors:
94

APPENDIX A. ERROR ESTIMATION OF PIV EXPERIMENTAL DATA 95
1. As N increases, the interval gets narrower from the
√
N term. Thus, the only way to obtain
more precise estimates for the mean velocity, it is to increase the sample size.
2. The larger the sample standard deviation, the larger the confidence interval. This simply
means that noisy data, i.e., data with a large standard deviation, are going to generate wider
intervals than data with a smaller standard deviation.
The measured standard deviation was about 0.03 m/s and the sample size in most experiments
was 100 PIV realizations (i.e., 200 images for the two dimensional PIV and 400 images for the
stereoscopic PIV measurements). The confidence limit for the mean velocity is calculated for 95%
(i.e., α = 0.05 and tα/2,N−1 = 1.984) to be:
± U =
1.984 · 0.03
√
100
= ±0.005952m/s (A.3)
A.2 Confidence limits of standard deviation
Similarly, the theoretical error estimate for the standard deviation (and kinetic energy as presented
by Grant and Owens (1990) for relatively low turbulent intensity flows) is given by χ2
-test (Snedecor
and Cochran, 1989). The two-sided version tests against the alternative that the true standard
deviation is either less than or greater than the specified value:
s
(N − 1)s2
χ2
α/2,N−1
≤ σ ≤
s
(N − 1)s2
χ2
1−α/2,N−1
(A.4)
where χ2
1−α/2,N−1 is the critical value of the χ2
-distribution with N − 1 degrees of freedom. For the
confidence level of 95% and N = 100 samples, the critical values are 128.422 and 73.366, respectively.
Thus, the standard deviation, that was estimated to be 0.03 m/s is limited between 0.0263 m/s and
0.0348 m/s, or in other words, the confidence interval of the standard deviation is ±0.004 m/s.
Furthermore, for relatively low turbulent intensity values, that is smaller than 30%, Grant and
Owens (1990) have shown that the error in estimation of turbulent intensity is well approximated
by the error in the standard deviation, presented here.

A.3 Error estimation at the measured velocity from the PIV
In this section we should derive the analytical base of the estimation of the velocity measurement
error in the PIV algorithm. The velocity vector in each interrogation area is calculated from the
pixel displacement, x:
u =
x
t
· M (A.5)
t is the time delay between the two laser pulses, and M is the scaling factor between the real object
size in the laser sheet plane and the size of the object image in the image plane (i.e., spatial scale
factor). The velocity error is estimated as:
∆v
v
=
∆x
v
∂v
∂x
+
∆M
v
∂v
∂M
+
∆t
v
∂v
∂t
(A.6)
where ∆x is the error, controlled by the particle density, the distribution of particles size, and the
error included in the sub-pixel displacement fitting (for example, Gaussian or parabolic shape fitting).
This ∆x error of the pixel displacement, is estimated by Keane and Adrian (1990), Westerweel (1997)
to be of order O(0.05) pixel. This estimation is based on the Gaussian two dimensional fitting of
the sub-pixel displacement, but with the condition that ∆x will less or equal to the quarter of the
size of the interrogation area (e.g., ≤ 8 pixels for the 32×32 pixel interrogation block) (according to
Adrian, 1991). The error of the time separation, ∆t is estimated as one thousand of the time delay
between the two laser pulses, ∆t ≈ 1 − 3 µs. The error of the scaling factor ∆M is calculated in the
following equation:
∆M = ∆SO
∂M
∂SO
+ ∆SI
∂M
∂SI
≈ 0.96 × 10−6 m
pixel
(A.7)
where SO - size of the object (i.e., particle) in the laser sheet plane (i.e., focus plane), and SI - size
of the object in the imaging plane, ∆SO - possible error of the object size ≈ 1 mm, and ∆SI - error
of the image size ≈ 1/2 pixel.
By using the following experimental conditions for the error estimation calculation: streamwise
velocity of 0.3 m/s, time delay between pulses t = 3000 µs, pixel displacement of order O(8) pixels,
and the scaling factor is M = 100 µm/pixel, we calculate the velocity error by equation (A.6) to be
∼ 1.2%. This result is in a good agreement with the common PIV accuracy analysis, presented by
Adrian (1991), Raffel et al. (1998), Ullum et al. (1998).
The above error estimation analysis, however, has not include the rejection case, where the
velocity vector is rejected as an outlier (e.g., during the filtering stage), either as a biased error

at the correlation part of the velocity calculation (for example due to low seeding density in a
particular interrogation point). On average, the number of rejected vectors in the PIV measurements
for different experimental conditions and configuration is O(10)%, when the number arises as one
approaches the the near-wall region of the flume.

Appendix B
PIV validation
In this section, the validation procedure with the aim to check the accuracy of the PIV measurements
is presented and following approaches are discussed in details:
1. Software performance analysis versus self-made PIV algorithm,
2. Integral comparison of the velocity data versus volumetric flow rate measurements,
3. Velocity measurements comparison versus Laser Doppler Velocimetry (LDV) results,
4. Out-of-plane velocity component validation by using two different experimental setups.
B.1 Flow rate comparison
First step of the validation is the integral approach where the flow rate calculated on the basis of the
average velocity profile and integration over the rectangular cross-section of the flume is compared
versus the flow rate meter of a V-cone type. The flow meter is installed on the supplying pipeline, as
shown in figure 5.1, after the pump, and measures the flow rate in liter-per-minute (lpm) units. The
measurement error according to the catalog is 0.5% of the scale. The following table B.1 presents
the comparative results of the flow rates. The estimated flow rate is always overestimated due to
the fact that the velocity values near the side walls are lower than the average velocity profile, which
is measured at the middle plane of the flume in x1 − x2 configuration (i.e., x3 = 15cm).
98

APPENDIX B. PIV VALIDATION 99
Case no. Q (lpm) U∞ [m/s] Qest Error, (%)
1 280 0.20 288 2.8
2 360 0.26 374 3.8
3 525 0.38 547 4.1
Table B.1: Comparison of the flow rate estimated by PIV measurements and directly measured by
flowmeter
.dwitq cn ir dcecne PIV -d ze‘vez jezn zkxreynd dwitqd oia d‘eeyd :a.1 dlah
B.2 Software performance
In order to detect possible bias error inherently introduced by some PIV algorithms, there are
two possible options (i) to analyze standard, synthetically produced images, introduced to the
PIV community by Okamoto et al. (2000), (ii) analyze the real PIV images by different soft-
ware packages. All the PIV measurements, in two dimensional, in stereoscopic and in multiplane
modes are performed by using a commercial software package InsightTM
(Inc., 1999a) that is a
part of the SPIV system. Figure B.1 presents the view of the Insight analysis of the standard
images used in the first-type evaluation analysis. The analysis is performed for the case no. 1
(http://piv.vsj.or.jp/piv/java/image01-e.html) and the evaluation shows an excellent agree-
ment between the known displacement of 7.5 ± 3 pixels and the InsightTM
analysis results in fig-
ure B.1.
The second type evaluation comparison is performed by using the URAPIV, self-made evaluation
software, developed with Matlab r
(http://urapiv.tripod.com). This software allows to analyze
PIV images by using the FFT-based cross-correlation calculation and includes the most common
velocity filters. The advantage of the URAPIV package is its open source and control of all possible
parameters. Figure B.2 depicts a comparison between the results of two software packages. The
graph includes the pixel displacements of the water flow, seeded with hollow glass sphere particles,
and evaluated by using the cross-correlation algorithm. The difference between two results is only
at the level of sub-pixel displacements and is less that 1%.

Figure B.1: Standard PIV image and the analysis with InsightTM
software
InsightTM
-d zpkez mr dfilp‘e ziihpiq zpenz :a.1 xei‘
Figure B.2: Comparison of the results calculated by InsightTM
and URAPIV software.
.URAPIV -e InsightTM
zepkez mr aeyign ze‘vezd z‘eeyd :a.2 xei‘

B.3 Comparative measurements with PIV and LDV systems
The PIV velocity measurements are validated by comparison with LDV results in the different
experimental setup, shown in figure B.3. The reason is that the LDV system is located at the
Danciger Labs and the experiments were performed in the instructional laboratory. The LDV
system is a commercial system from Dantec Inc., based on the He-Ne laser (20 mW, 632.8 nm),
collimator, expander, mirror and the photodetector (10 kHz acquisition frame rate) shown in the
figure B.3. The PIV system measured the in-plane velocity component (from the left to right in
the measurement plane), and the LDV measurement volume was located in an appropriate way to
measure the same velocity component.
The chosen configuration is a small glass cylinder (r = 8 cm) with the swirling water flow
(h = 10cm), originated by the DC stirrer. The velocity presented in this study is the radially
directed profile (from the center toward the side wall of the cylinder). The PIV camera was located
above the cylinder during the PIV experiment and the laser sheet was located 6 cm from the bottom
of the cylinder. The PIV results are of the ensemble average of 200 images (i.e., 100 velocity fields),
analyzed with 64×64 interrogation grid and overlapping of 50%, time separation of 1000 µs. During
the LDV measurements, the measurement volume was located at several radial locations (10 points)
at the same height, and the results of the averaged 100 samples are depicted in figure B.4. The
results present good agreement between the two profiles, with the mean relative difference of 2.3%.
The x-axis of this graph is the radial coordinate (r) and the y-axis is the measured velocity, U.
Presence of the strong shear in the center, and excessive light scattering near the cylinder walls due
to the curvature of the cylinder, reduced the accuracy of the LDV measurements and omitted from
the presented graph. The comparison shows the accurate validation of the PIV system, comparable
to the known validation techniques (Raffel et al., 1998).
B.4 Out-of-plane component validation
In contrast to the high accuracy of the two dimensional PIV measurements (i.e., within ± 2% limit),
the out-of-plane component, measured by stereoscopic PIV is reported to be of order O(8%) (Raffel
et al., 1998). In order to validate our specific system, its calibration and evaluation software, we
compare the out-of-plane component of the velocity in x2 − x3 mode, i.e., streamwise velocity U1

Figure B.3: Schematic view of the PIV and LDV measurement systems and flow configuration.

Figure B.4: Velocity results of the PIV versus LDV measurement results.
with the in-plane two dimensional measurements in x1 − x2 experimental configuration (see 5.5.
The comparison is on the average profile level, that is provides a qualitative accuracy estimation,
based on two sets of 100 vector maps, taken at the middle x1 − x2 plane and the x2 − x3 plane.
Figure B.5 presents the average streamwise velocity profile U1(x2) versus the distance from the wall.
The number of measurement points in the out-of-plane mode is smaller due to the smaller imaging
field of view in this configuration. The depicted result proposes that our SPIV system is compatible
with the published (Soloff et al., 1997) error level of 7 − 8%.
B.5 Velocity derivatives, calculation and validation

Figure B.5: Velocity profile measured in x − y (+) and in y − z (•) configurations.

Appendix C
Derivatives. Part 1: Vorticity
calculation
C.0.1 Standard differentiation schemes
Since PIV provides the velocity vector field sampled on a two- or three-dimensional, usually evenly
spaced grid, finite differencing has to be employed in the estimation of the spatial derivatives of
the velocity gradient tensor, dU/dX. Moreover, the velocity data is disturbed by noise, that is, a
measurement uncertainty, U . Although the error analysis used for the estimation of the uncertainty
in the differentials assumes the measurement uncertainty of each quantity to be decoupled from
its neighbors, this is not always the case. For instance, if the PIV image is oversampled, that is,
the interrogation interval (sample points) is smaller than the interrogation area dimensions, the
recovered velocity estimates are not independent because the neighboring interrogation areas partly
sample the same particles. For simplicity the differentiation schemes described next it is common
to assume the measurement uncertainties to be independent of their neighbors.
Table C.1 lists a number of finite difference schemes to obtain estimates for the first derivative,
df/dx, of a function f(x) sampled at discrete locations fi = f(xi). The ’accuracy’ in this table
reflects the truncation error associated with derivation of each operator by means of Taylor series
expansion. The actual uncertainty in the differential estimate due to the uncertainty in the velocity
estimates u can be obtained using standard error propagation methods assuming the individual
105

APPENDIX C. DERIVATIVES. PART 1: VORTICITY CALCULATION 106
data to be independent of each other.
The difference between the Richardson extrapolation scheme and the least squares approach, is
that the former is designed to minimize the truncation error while the latter attempts to reduce the
effect of the random errors, that is, the measurement uncertainty, U . The least squares approach
therefore seems to be the most suitable method for PIV data. In particular, for oversampled velocity
data where neighboring data are no longer uncorrelated, the Richardson extrapolation scheme along
with the less sophisticated finite difference schemes will perform poorly with respect to the least-
squares approach. On the other hand, the least-squares approach has a tendency to smooth the
estimate of the differential because the outer data fi±2 are more weighted than the inner data
fi±1. The effect of oversampling on the estimation of the differential quantities could be analysed
as follows: by doubling the interrogation window overlap, much noisier vorticity fields are obtained
due to two related causes: first the grid spacing, ∆x, ∆y is reduced by a factor of two while the
measurement uncertainty for the velocity, U , stays the same. As a result the vorticity measurement
uncertainty is doubled. Secondly, all or part of the data used in the differentiation scheme will be
correlated because of the increased overlap. For instance velocity gradient induced bias errors will
be similar in neighboring points which in turn results in a biased estimate of the vorticity. Thus, the
estimation of differential quantities from the velocity field has to be optimized with respect to the
grid spacing. A coarser grid yields less noisy estimates of the gradient quantity, but also results in a
reduced spatial resolution. In the following section alternative differentiation schemes are introduced
which perform well even in oversampled data.
C.0.2 Alternative differentiation schemes
The finite differencing formulae given in table C.1 have been derived for functions of one variable,
that is, they are applied in one dimension at a time. The velocity data obtained by PIV is provided
on a two- or three-dimensional grid which also holds for the differential quantities obtained from
where it. As a consequence the use of one-dimensional finite difference schemes for the estimation
of the two-dimensional differential field quantities seems inadequate.
By definition the vorticity is related to the circulation by Stokes theorem
Γ =
I
U · dl =
Z
(∇ × U) · dS =
Z
ω · dS (C.1)

Operator Implementation Accuracy
Forward difference

df
dx

i+1/2
≈ fi+1−fi
∆x O(∆x)
Backward difference

df
dx

i−1/2
≈ fi−fi−1
∆x O(∆x)
Center difference

df
dx

i
≈ fi+1−fi−1
∆x O(∆x2
)
Forward difference (3
points)

df
dx

i+1/2
≈ −3fi+4fi+1−fi+2
2∆x O(∆x2
)
Backward difference (3
points)

df
dx

i−1/2
≈ fi−2−4fi+3fi
2∆x O(∆x2
)
Least squares

df
dx

i
≈ 2fi−2+fi+1−fi−1−2fi−2
10∆x O(∆x2
)
Forward/Backward
difference (5 points)

df
dx

i±1/2
≈ −25fi+48fi+1−36fi+2+16fi+3−3fi+4
12∆x O(∆x4
)
Richardson
extrapolation

df
dx

i
≈ fi−2−8fi−1+8fi+1−fi+2
12∆x O(∆x4
)
Table C.1: First order differential operators for data spaced at uniform intervals
where l describes the path of integration around a surface S. The vorticity for a fluid element is
found by reducing the surface S, and with the path l, to zero:
ˆ
~
n · ω = ˆ
~
n · ∇ × U = lim
S←0
1
S
I
U · dl (C.2)
where the unit vector ˆ
~
n is normal to surface S. Stokes theorem can also be applied to the PIV
velocity data (two-dimensional):
(ω̄z)i,j =
1
A
Γi,j =
1
A
I
l
(U, V ) · dl (C.3)
where (ω̄z)i,j reflects the average vorticity within in the enclosed area. In schemes for practice
equation (C.3) is implemented by choosing a small rectangular contour around which the circulation
is calculated using a standard integration scheme as the trapezoidal rule. The local circulation s then
divided by the enclosed area to arrive at an average vorticity in this area. The following formula
provides a vorticity estimate at point (i, j) based on a circulation estimate around the neighboring
eight points:
(ω̄z)i,j ≈
Γi,j
4∆x∆y
(C.4)

Γi,j =
1
2
∆x(ui−1,j−1 + 2ui,j−1 + ui+1,j−1) (C.5)
+
1
2
∆y(vi+1,j−1 + 2vi+1,j + vi+1,j+1)
+
1
2
∆y(vi+1,j−1 + 2vi+1,j + vi+1,j+1)
−
1
2
∆x(ui+1,j+1 + 2ui,j+1 + ui−1,j+1)
−
1
2
∆y(vi−1,j+1 + 2vi−1,j + vi−1,j−1)
An inspection of equation (C.4) reveals that the expression is equivalent to applying the center
difference scheme to a smoothed (3 × 3 kernel) velocity field. While the vorticity estimation by
one-dimensional finite differences requires only 4 to 10 velocity data values this expression utilizes
12 data values. The uncertainty in the vorticity estimate, assuming the uncorrelated velocity data,
then reduces compared to center differences or Richardson extrapolation method. Also the effects
due to the data oversampling are not as significant because no differences of directly adjoining data
are used.
A similar approach may be used in the estimation of the shear strain and the out-of-plane strain:
(xy)i,j =

∂u
∂y
+
∂v
∂x

i,j
= −
ui−1,j−1 + 2ui,j−1 + ui+1,j−1
8∆y
(C.6)
+
ui+1,j+1 + 2ui,j+1 + ui−1,j+1
8∆y
−
vi−1,j+1 + 2vi−1,j + vi−1,j−1
8∆x
+
vi+1,j−1 + 2vi+1,j + vi+1,j+1
8∆x
.
(zz)i,j =

∂u
∂x
+
∂v
∂y

i,j
=
vi−1,j−1 + 2vi,j−1 + vi+1,j−1
8∆y
(C.7)
−
vi+1,j+1 + 2vi,j+1 + vi−1,j+1
8∆y
+
ui+1,j−1 + 2ui+1,j + ui+1,j+1
8∆x
−
ui−1,j+1 + 2ui−1,j + ui−1,j−1
8∆x
.
C.0.3 Uncertainties and errors in differential estimation
As already noted a variety of factors enter in the uncertainty of a differential estimate.

Uncertainty in velocity: Each PIV velocity estimate Ui,j is associated with a measurement un-
certainty u whose magnitude depends on a wide variety of aspects such as interrogation
window size, particle image density, displacement gradients, etc. Since differential estimates
from the velocity data require the computation of local differences on neighboring data the
noise increases inversely proportional to the local difference as the spacing between the data
neighboring points is reduced. That is, the estimation uncertainty in the differential, ∆, scales
with u/∆x.
Oversampled velocity data: It is common practice to oversample a PIV recording during interro-
gation at least twice to fulfill the Nyquist sampling theorem as well as to bring out small-scale
features in the flow. Because of this oversampling, neighboring velocity data are estimated
partially from the same particle images and therefore are correlated with each other. Because
of this, neighboring data axe likely to be biased to a similar degree, especially in regions con-
taining high velocity gradients and/or low seeding densities. This localized velocity bias then
causes the differential estimate to be biased as well. The oversampling effects can be observed
in simulation results.
Interrogation window size: The size of the interrogation window in the object plane defines the
spatial resolution in the recovered velocity data, provided the sampling positions fulfill the
Nyquist criterion. The spatial resolution in the velocity field in turn limits the obtainable
spatial resolution of the differential estimate. Depending on the utilized differentiation scheme
the spatial resolution will be reduced to some degree due to smoothing effects.
Curvature effects: The standard PIV method only is a first order approximation to the true
particle image displacement. Because it generally relies on only two illumination pulses, effects
due to acceleration and curvature are lost. In regions of rotation (i.e. velocity gradients) this
straight line approximation underestimates the actual particle image displacement and thereby
the local velocity. Differential estimates will then have a tendency to be biased to lower
magnitudes as well. By reducing the illumination pulse delay, ∆t, this effect can be reduced
at the cost of increased noise in the differential estimate due to the velocity measurement
uncertainty, u, itself.

C.1 Implementation
C.1.1 The test case
The instantaneous flow field of the Taylor-Green vortex flow is a good choice to evaluate the vorticity
and derivatives calculation algorithms because it includes strong rotation. In our research, the
rotation component is the most encountered object and interesting quantity since we are going to
investigate vortex structures and turbulent coherent structures through PIV measurements. The
flow field is given by following equations in two dimensions and satisfies the equation of continuity:
u = Wsin

2πkx
L

· cos

2πky
L

(C.8)
v = −W cos

2πkx
L

· sin

2πky
L

(C.9)
where u and v stand for velocity components in x and y-directions, respectively, W is the maximum
amplitude of the two velocity components, k is the wave number, L is the width of the square
domain, x and y are Cartesian coordinates in the square domain. The vorticity of such field is
computed analytically and it is equal to
ωz = 4
Wπk
L
sin

2πkx
L

sin

2πky
L

(C.10)
The flow field is shown in Figure C.1 with velocity arrows and a background color that represents
the vorticity magnitude.
C.1.2 Numerical error estimation
In order to estimate the error of different numerical differentiation schemes, we use the idea similar
to the Monte-Carlo simulation: the numerically produced random ’noise’, normally distributed with
different levels of the standard deviation is added to the known velocity field. The noise is added
separately to each of the velocity components. The procedure is repeated several times for different
number of simulation runs (i.e. number of generated velocity maps with the added noise). For each
artificially generated velocity map we calculate the vorticity field, using one of the described above
differentiation methods and then the error is estimated between the estimated mean vorticity field
and the exact vorticity data. The error propagation due to the different number of simulation runs

Figure C.1: Taylor-Green vortex flow field and its vorticity.
that represent the number of velocity maps in real PIV ’batch’ test, and/or due to the several levels
of the additive noise is calculated and presented here.
Implementation of the differentiation schemes Differentiation schemes applied to the velocity
maps in two-dimensional case is implemented using the computational effective 2D convolution
method (for more details see FASTDERIVATIVE.M and VORTICITY.M procedures).
Numerical simulation The numerical simulation procedure is performed using Matlab r
NORMRND
function. The mean value (µ) is set to zero, and standard deviation (σ) is set so the maximum
velocity uncertainty takes a value of 2, 5, or 7.5% of the maximum velocity in the original
average field. The simulation performed with 100, 500, and 1000 simulation points. Therefore
9 values of the error estimation is calculated with the described numerical procedure. The
results of the simulation and concluded remarks on the performance of differentiation schemes
are presented below

Results
The results are presented in a graphic way with a help of Figures C.2 and C.3. Figure C.2 presents
the relative error of the vorticity estimation (relative to the exact vorticity field) as a measure of the
vorticity uncertainty versus the number of simulation runs. Three plots (i.e. (a), (b) and (c)) are
for different levels of the additive noise (i.e., 2, 5, and 7.5%, respectively). The second figure (Figure
C.3) is the ’other side projection’ of the same results and shows the propagation of the relative error
with raising levels of the error, while different plots (i.e. (a), (b) and (c)) are for different number
of simulation runs (i.e., 100, 500, and 1000 matrices, respectively).
Figure C.2: Relative error as a function of simulation runs number for (a) 2%, (b) 5% and (c) 7.5%
velocity noise level.
Figures C.2 and C.3 shows that in most cases, the ’least squares’ and ’circulation’ methods are
the most robust methods that are less dependent on the number of PIV velocity maps available
for the vorticity calculation and on the velocity uncertainty level. This result could be expected,
according to the previous explanation (see SectionC.0.1). Both methods uses smoothed velocity data
and provides more weight to more neighbor points. The ’circulation’ method is seems to be more
attractive, due to the fact that it uses 12 neighbor points instead of 8 in other methods but it lacks

Figure C.3: Relative error as a function of additive noise level for (a) 100, (b) 500, and (c) 1000
simulation runs.
the ability to get derivatives for other vortex eduction methods separately. The decision becomes
much less clear when the number of simulation runs or, in other words, number of PIV maps is
relatively small (about 100 – 200 maps). In such case, the ’Richardson’ method is more attractive,
probably due to its ability to treat more effectively in the different type of errors. It is no clear when
and why ’higher-order’ differentiation schemes (’Richardson’and ’Least Squares’) are better than the
conventional 3-point central difference scheme, which in turn, requires less computing resources, but
there is overall impression that higher-order schemes performs better in noisy environment such as
real PIV measurements.
The continuity test
The relative error analysis presented here does not show the independence between the derivatives
errors and the velocity values and, in addition, the leaves the question about spatial randomness of
the derivatives schemes. In order to answer these questions, the derivatives used in the calculation
of the continuity equation, or in other words, the estimation of the out-of-plane strain zz (Eq.
refeq:epsilonzz). The results of this simulation are difficult to present in few graphs, so we decided
to show an example in the Figure reffig:continuity, that shows the colour contour maps of the mean

values of the η. The simulation presented in the Figure reffig:continuity is a result of 500 runs with
an noise level of 5%. The colorbar below each one of four plots shows the scale of that graph and
the title above presents the calculation scheme used for the calculation. It is easily seen that the
upper row of ’Center’ and ’Richardson’ schemes contains of much larger values of the error, than
the lower row of ’Least Squares’, and ’Circulation’ schemes. Clearly from all plots, that there is an
error in the calculation, due to the fact, that analytical flow field, chosen for this evaluation, has
a zero out-of-plane strain (zz = 0). It is also clear that the error origins from the simulation rule
of the additive white noise. The most important conclusions from the analysis of this part of the
simulation are listed below:
• The differentiation schemes are independent of the velocity field values (i.e., random spatial
distribution of the error);
• The mean error values do not propagate as the additive white noise level rises;
• The ’Circulation’ method (i.e. integral method) is the best for calculating the out-of-plane
strain estimation, but leaves the necessity of the derivative calculations for several eduction
schemes.
• The ’Least Squares’ method, that is based on direct calculations of the derivatives in all
directions and consequential calculation of different estimations seems to be the most optimized
selection at this stage of the analysis.
Figure C.4: Mean value of the out-of-plane strain estimation (the mean error of the continuity
equation) for the 500 simulation runs and 5% additive noise level. The plot consists of the results
for the ’Center’ - upper left, ’Richardson’ - upper right, ’Least Squares’ - lower left, and
’Circulation’ calculation scheme at the lower right corner.
C.1.3 Conclusions
As it is described above, two differentiation schemes stay candidates for further research based on
the derivatives, i.e., the discriminant analysis, the vorticity-based conditional sampling, etc. Those
schemes are ’Circulation’ (or ’Integral’) and the ’Least Squares’ schemes. While former (or actually
two very similar schemes for the vorticity and for the out-of-plane strain estimations) is the integral

based scheme, which uses 12 (or more) neighbor velocity vectors for the parameter estimation seems
to provide the most accurate and robust results over whole simulation set, it does not provide us with
derivatives. If the analysis is a discriminant-based analysis, or in other words, the kinematic flow
analysis, we have, in addition to the vorticity or out-of-plane strain, to calculate the full gradient
tensor. Therefore, in order to avoid the multiple calculations of the same values, the ’Least Squares’
method which uses 8 neighbor data points for the direct derivative estimations, provides us both
with quite accurate and robust results and complete gradient tensor of the flow.

C.2 Appendix B - Impinging Jet Test
In order to apply, at least partially, chosen differentiation schemes on realistic PIV images, previously
presented experimental data from the air impinging jet measurements have been used as a test case.
The measurement set consists of 135 pairs-of-images, that is, 135 velocity maps. The number of
the images is a result of the velocity r.m.s. divergence analysis (i.e., the number of velocity maps
after that the r.m.s. velocity values do not change). In previous research vorticity field have been
calculated with the simplest, three-point center difference differentiation scheme (entitled ’Center’
in this work). Unfortunately, the ’average’ test showed some small, but an error, or in other words,
the average of the 135 instantaneous vorticity maps was not exactly equal to the vorticity field,
calculated from the average velocity field. In order to reproduce that analysis, we calculated the
same values again with least square differentiation scheme and the results shown in Figure C.5.
where the actual vorticity map is presented in Figure C.6.
Figure C.5: Difference between the average of vorticity fields and the vorticity of the average
velocity field.
Figure C.6: Average vorticity field of the impinging jet flow.
Additional calculation that could be shown here is of the out-of-plane strain estimation. We do

not expect to get the zero values over the whole field, but after the complete test of the derivatives
calculation, we can base our conclusions with much larger certainty on the accepted results. The
following Figure C.4 shows that except the region of the air-plate impinging, the flow is almost
completely in-plane and the out-of-plane component is approximately zero. At the impinging region,
the out-of-plane components are more significant and it is probably most due to the asymmetry of
the experimental apparatus.

C.3 Appendix B - Matlabr
procedures
FASTDERIVATIVE.M
function [varargout] = fastderivative(varargin)
%FASTDERIVATIVE Calculates fast derivatives of two-dimensional data
% using 2D convolution (See conv2 for more info).
% [DFDX,DFDY] = FASTDERIVATIVE(F,HX,HY,’METHOD’)
% F - 2D matrix, HX,HY - intervals in X,Y directions.
% ’method’ - one of the following methods:
% ’leastsq’ - 5 points
% ’richardson’ - 5 points
% ’center’ - 3 points, center difference
% ’general’ - default, forward/backward/center in one
% ’5point’ - richardson plus forward/backward on boundaries
%
% [DFDX,DFDY] = FASTDERIVATIVE(F,H,’METHOD’)
% Uses HY = HX = H,
%
% [DFDX,DFDY] = FASTDERIVATIVE(F,’METHOD’)
% Uses default HX,HY = 1.
% Created: 05-Mar-2001
% Author: Alex Liberzon
% E-Mail : liberzon@tx.technion.ac.il
% Phone : +972 (0)48 29 3861
% Copyright (c) 2001 Technion - Israel Institute of Technology
%
% Modified at: 05-Mar-2001
% $Revision: 1.0 $ $Date: 05-Mar-2001 11:59:07$
% Parse inputs
if nargin == 4
[f,hx,hy,method] = deal(varargin{:});
elseif nargin == 3
[f,hx,method] = deal(varargin{:});

hy = hx;
elseif nargin == 2
[f,method] = deal(varargin{:});
hx = 1; hy = 1;
end;
switch lower(method)%
case ’general’,’default’,’3points’,’threepoints’}
dfdx = -conv2([0 1 0],[-1 0 1],f,’same’)/2/hx;
dfdy = -conv2([-1 0 1],[0 1 0],f,’same’)/2/hy;
dfdx(:,1) = f(:,2)-f(:,1);
dfdx(:,end) = -f(:,end-1)+f(:,end);
dfdy(1,:) = f(2,:)-f(1,:);
dfdy(end,:) = -f(end-1,:)+f(end,:);
case {’leastsq’}
dfdx = -conv2([0 0 1 0 0],[-2 -1 0 1 2],f,’valid’)/10/hx;
dfdy = -conv2([-2 -1 0 1 2],[0 0 1 0 0],f,’valid’)/10/hy;
case {’center’}
dfdx = -conv2([0 1 0],[-1 0 1],f,’valid’)/2/hx;
dfdy = -conv2([-1 0 1],[0 1 0],f,’valid’)/2/hy;
case {’richardson’}
dfdx = -conv2([0 0 1 0 0],[1 -8 0 8 -1],f,’valid’)/12/hx;
dfdy = -conv2([1 -8 0 8 -1],[0 0 1 0 0],f,’valid’)/12/hy;
otherwise
error(’Wrong method, see HELP FASTDERIVATIVE ’);
end
varargout{1} = dfdx;%
varargout{2} = dfdy;%
VORTICITY.M
function [varargout] = vorticity(varargin)
%VORTICITY Calculates vorticity and checks the continuity equation.
% [OMEGA,EZZ] = VORTICITY(X,Y,U,V,METHOD)
% returns the vorticity OMEGA and Z-component of the strain EZZ.

%
%
%
% See also FASTDERIVATIVE, PROCS_V4
%
% Created: 02-Mar-2001
% Author: Alex Liberzon
% E-Mail : liberzon@tx.technion.ac.il
% Phone : +972 (0)48 29 3861
% Copyright (c) 2001 Technion - Israel Institute of Technology
%
% Modified at: 05-Mar-2001
% $Revision: 1.02 $ $Date: 05-Mar-2001 23:12:01$
% Inputs:
% The simplest case
[x,y,u,v,method] = deal(varargin{:});
% Parameters:
% Homogeneous grid spacing - the simplest case
DeltaX = max(abs(diff(x(1:2,1))),abs(diff(x(1,1:2))));%
DeltaY = max(abs(diff(y(1:2,1))),abs(diff(y(1,1:2))));%
switch lower(method)%
case {’circulation’,’circ’}
dx=(1/(8*DeltaX)).*...
[-1 0 1
-2 0 2
-1 0 1];
dy=(1/(8*DeltaY)).*...
[-1 -2 -1
0 0 0
1 2 1];
vort = - conv2(v,dx,’valid’) + conv2(u,dy,’valid’);
dvdy = conv2(v,dx,’valid’);

dudx = conv2(u,dy,’valid’);
cont = dudx + dvdy;
case {’richardson’}
[dudx,dudy] = fastderivative(u,DeltaX,DeltaY,’richardson’);
[dvdx,dvdy] = fastderivative(v,DeltaX,DeltaY,’richardson’);
vort = dvdx - dudy;
cont = dudx + dvdy;
case {’leastsquares’,’leastsq’}
[dudx,dudy] = fastderivative(u,DeltaX,DeltaY,’leastsq’);
[dvdx,dvdy] = fastderivative(v,DeltaX,DeltaY,’leastsq’);
vort = dvdx - dudy;
cont = dudx + dvdy;
case {’center’}
[dudx,dudy] = fastderivative(u,DeltaX,DeltaY,’center’);
[dvdx,dvdy] = fastderivative(v,DeltaX,DeltaY,’center’);
vort = dvdx - dudy;
cont = dudx + dvdy;
otherwise
disp(’Wrong method’);
end
varargout{1} = vort; varargout{2} = cont;
Simulation procedure: PROCS V4.M
clear all%
[x,y,u,v,vort,dU] = vortex_flow(1,1,1,0);%
indx = 0; indy = 0;
for N = [100,500,1000]
indx = indx + 1;
indy = 0;
for errorlevel = [0.02 0.05 0.075]
indy = indy + 1;
U = zeros([size(u),N]);
V = zeros([size(u),N]);

% Monte Carlo:
for i = 1:N,
U(:,:,i) = u + normrnd(0,errorlevel*max(u(:))/1.96,[size(u)]);
end
for i = 1:N,
V(:,:,i) = v + normrnd(0,errorlevel*max(v(:))/1.96,[size(v)]);
end
[omegar,contr,omegal,contl] = deal(zeros(47,47,N));
[omegac,contc,omega3,cont3] = deal(zeros(49,49,N));
for i = 1:N,
[omegar(:,:,i),contr(:,:,i)] = vorticity_v4(x,y,U(:,:,i),V(:,:,i),’richardson’);
[omega3(:,:,i),cont3(:,:,i)] = vorticity_v4(x,y,U(:,:,i),V(:,:,i),’center’);
[omegal(:,:,i),contl(:,:,i)] = vorticity_v4(x,y,U(:,:,i),V(:,:,i),’leastsq’);
[omegac(:,:,i),contc(:,:,i)] = vorticity_v4(x,y,U(:,:,i),V(:,:,i),’circulation’);%
end;
% End of calculations
% Relative error = (mean(omegaI(:,:,i),3) - vort)/rms(vort);
relerror3(indx,indy) = rms((mean(omega3,3)-vort(2:end-1,2:end-1))/...
rms(vort(2:end-1,2:end-1)));
relerrorc(indx,indy) = rms((mean(omegac,3)-vort(2:end-1,2:end-1))/...
relerrorr(indx,indy) = rms((mean(omegar,3)-vort(3:end-2,3:end-2))/...
relerrorl(indx,indy) = rms((mean(omegal,3)-vort(3:end-2,3:end-2))/...
end% End of errorlevel loop
end% End of N loop
% Continuity check
figure,subplot(221),
pcolor(mean(cont3,3)),shading interp, colorbar horiz
title(’3 point, center difference’);
subplot(222),pcolor(mean(contr,3)),shading interp, colorbar horiz

title(’Richardson’)
subplot(223),pcolor(mean(contl,3)),shading interp, colorbar horiz
title(’Least Squares’)
subplot(224),pcolor(mean(contc,3)),shading interp, colorbar horiz
title(’Circulation’)

Appendix A
XPIV - Image processing definitions
A.1 Percentile
The percentile, p (in percents, p%), of a gray level intensity distribution (P(I)) is defined as the
intensity level n that p percents of the pixels has the gray level intensity less or equal to n. We
recognize three known cases of the percentiles: 0% is the minimum gray level value in the image I,
50% is the median intensity level of the specified histogram and 100% is the maximum value.
A.1.1 Histogram based operations
In PIV experiments it is difficult to make use of the full dynamic range of the camera. Therefore,
PIV images often have the minimum and maximum values (i.e., 0% and 100%) that are not equal
to 0 or 255 (for 8 bit images). One can improve the images by stretching the intensity distribution
(histrogram) over the desired dynamic range. Within the XPIV algrorithm the histogram stretching
is applied to achieve the maximum available range:
J(m, n) = 255 ·
I(m, n) − 0%
100% − 0%
, (A.1)
where 0% and 100% are minimum and maximum gray level intensity values as defined by percentiles
above.
124

APPENDIX A. XPIV - IMAGE PROCESSING DEFINITIONS 125
A.1.2 Derivative based operations
Since digital image processing techniques deal with discrete digitized images presented over the
rectangular grid, I(m, n), the derivative based operations lie on the derivative approximations. Thus
the image gradient ∇I is approximated by the sum of two convolutions in row-wise and column-wise
directions, denoted as x and y, respectively:
∇I = (hx ⊗ I)x + (hy ⊗ I)y
where subscripts x and y denote the projections on the x or y axis. Thus, the gradient magnitude
is defined as:
Ig =
q
(hx ⊗ I)
2
+ (hy ⊗ I)
2
.
There are many possible operators for hx and hy and most simple ones are [1 − 1] or [1 0 − 1] as
hx and their transposes, hy = ht
x.
A.2 Morphology based operations
In morphology based operations, we refer to the image (either binary or intensity level) as a set of
objects and background (Serra, 1982, Giardina, 1988):
I = {x |I(x) ⊂ Y }
where Y is the set under R2
that satisfies some predefined property. For example, for the binary
image of 0’s and 1’s, objects are all pixels that are equal to 1, i.e. I = {x |I(x) = 1}. The background
or in other definition, complement of I is defined as all elements that are not in set of I:
Ic
= {x |x /
∈ I}
Each object can be treated by one of four basic operators: union ∪, intersection ∩, complement c
and translation, which is defined for set I and vector b as:
I + b = {x + b |x ∈ I}
or in other words, all pixels that are covered by set I (an object) by shifting, as it is defined in vector
b.
Two basic mathematical morphology operations for binary sets {0, 1} are defined as following:

Dilation:
I ⊕ B =
[
b∈B
(I + b)
Erosion:
I B =

b∈B
(I + b)
Usually, I is denoted as an ”object” or an ”image”, and B as a structuring element. From dilation
and erosion, one can define higher order operations:
Opening:
I ◦ B = (I B) ⊕ B
Closing:
I • B = (I ⊕ B) B
The morphology operations defined above for the binary images were extended to gray level
images (Serra, 1982) by substituting max operator for union ∪, and min for intersection ∩:
Dilation:
I(m, n) ⊕ B = max
{i,j}∈B
{I(m − i, n − j) + B(i, j)}
Erosion:
I(m, n) B = min
{i,j}∈B
{I(m − i, n − j) − B(i, j)}
In other words, for each pixel m, n the output of the dilation operation is the maximum of the
summation of a shifted original image I with all shifts i, j defined in B. The erosion are defined
analogically with the minimum and subtraction operators. Opening is defined as the erosion followed
by dilation, and closing is the opposite order of basic operations.
Using the basic operations we can define some filters and high-level image processing routines:
Smoothing:
˜
IB = ((I ◦ B) • B) , for large B

Gradient:
Ig =
1
2
((I ⊕ B) − (I B))
Laplacian:
∇2
= IL =
1
2
[((I ⊕ B) − I) − (I − (I B))] =
=
1
2
((I ⊕ B) + (I B) − 2I)
Background correction:
ˆ
IB = I − ˜
IB
A.3 Segmentation
Segmentation is one of the most common operations in high-level image processing and computer
vision. The target is to distinguish between the ”objects” and the ”background” and the most
popular techniques are thresholding and edge detection.
Thresholding:
IT = {x |I(x) ≥ T}
where T is the threshold. The threshold can be defined arbitrarily or by using one of the
common techniques defined on the histogram of the image. The isodata algorithm, for example,
is an iterative technique (Ridler and Calvard, 1978), where histogram is divided into two groups
by its mean value, and in the following iterations the threshold is defined as the average of
two mean values of each group, until the difference is insignificant. Another common and
fast technique is entitled background symmetry algorithm. This algorithm is based on the
assumption that the maximum peak within the histogram is of the background and uses
percentiles to define the limits of the background population. Then, using the symmetry
assumption, the threshold is defined as:
T = max {P(I)} − (p% − max {P(I)}) .

Edge detection: Finding pixels on the borders of the objects.
Two edge detection techniques have been used in the presented work:
Gradient-based:
Ie = {x ∈ I |Ig(x) ≥ T}
Zero-crossing: searching for the zero crossing of the Laplacian of the smoothed image:
Iz =
n
{x, y} IL

˜
IB

= 0
o

Appendix B
Surfactants
introduction
drag reduction effect in turbulent flow, by low concentrations of surfactants, is well known. the
changes could be spectacular with only few parts per million (ppm) of surfactant solution added to
the solvent (e.g., reviews in Ohlendorf et al., 1986, Zakin and Lui, 1983, Gyr and Bewersdorff, 1995).
the changes appear in number of flow characteristics, both of large and small scales. however, the
general belief is that surfactants, as other polymer additives, act directly toward the small scales
(see for references in Gyr and Bewersdorff, 1995, Tsinober, 2001).
the drag reduction effect is accompanied by the modification of the turbulence structure, such as
significant decrease of the reynolds stresses (Warholic et al., 1999) and turbulence production. the
suppression of reynolds stresses does not mean immediately that there is a significant reduction of
the energy of turbulent fluctuations (r.m.s. values). the turbulent energy in flows of drag reducing
solutions, have been shown to be sometimes smaller than in solvent flows, but also may be increased,
due to the fact that both turbulent energy production and the dissipation are reduced strongly in
the drag reducing flow (Tsinober, 1990). one of the surfactant effects that was reported by almost all
researches is the increased anisotropy when the wall-normal velocity fluctuations are considerably
suppressed. such observations have lead to the conclusion that the main reason for the reduced
reynolds stresses is the decorrelation of the streamwise (u1) and wall-normal (u2) components of the
velocity fluctuations (Tsinober, 2001).
129

APPENDIX B. SURFACTANTS 130
Figure B.1: schematic drawing of the experimental setup (top) and piv configuration (bottom)
in the present research we investigate experimentally the influence of the bio-degradable, non-
toxic surfactants from a class of alkyl polyglycosides. they are used in cosmetics and food industry,
and some of them can form rod-like micelles at relatively low concentrations. this makes their poten-
tial use as drag reducers attractive, although, to the best of our knowledge, they have not explored
yet. in section B we describe the experimental facility briefly and the particle image velocimetry
(piv) measurement parameters in more detail. section B presents the most interesting and important
results based on two-dimensional, two component velocity piv measurements in streamwise–spanwise
plane (i.e., parallel to the flume bottom wall). concluding remarks and summary of the main results
are given in section 7.
experimental facility
the investigated flow was in the turbulent boundary layer in a flume of 4.9 × 0.32 × 0.1 m. the flume
was filled with tap water and seeded with hollow glass spherical particles (average diameter of 10 µm)
for the first part of the experiment. the influence of the surfactant solution was measured with low
concentrated (20 parts per million) ”agrimul pg 2062” solution from the alkyl polyglycosides family
of bio-degradable, nonionic surfactants. a schematic diagram of the experimental configuration is
shown in figure B.1. this work includes the preliminary results based on the measurements at
reynolds number 20,000 based on the water height. the velocity field was measured 2.5 m from
the entrance and the field of view was 0.1 × 0.1 m in the streamwise-spanwise (x − z) plane. the
laser sheet was located 1 cm above the bottom of the flume, that is y+
≈ 80, according to our
previous results in the x − y plane, (Liberzon et al., 2001). a commercial piv system based on the
170 mj/pulse doubled nd:yag lasers and 1008 × 1008 pixels ccd camera, was utilized for this work.
by using 64 × 64 pixels square interrogation areas, a ≈ 100 µm/pixel ratio, and 50% overlapping,
the analysis produced about 1000 vectors in a given field of view. the statistical analysis was based
on 50 image pairs for the water and for the surfactant solution, resulting in 100 flow two-component
velocity vector maps.

results and discussion
in order to present the influence of the surfactant solution on the turbulent flow field, all the results
are presented in pairs of water and drag reduced flow, corresponding to the same flow rate, q,
or velocity uq. the reason is that the mean velocity of the flow increased from u ≈ 0.2 m/s to
≈ 0.25 m/s for the same flow rate due to the drag reducing effect of the surfactant solution.
the experiment was conducted for three different flow rates, namely low, intermediate and high
q = 300, 350, 400 liters per minute (lpm), uq = 0.15, 0.2, and 0.25 m/s, and re = 15, 000, 20, 000
and 25, 000, respectively. the evidences of the surfactant effect are shown for the intermediate case
(q = 350 lpm). the statistical significance of the effect is emphasized by the fact that all results
presented here are simple ensemble average denoted by h i.
figure 6.28 presents the comparison between the ensemble average of the streamwise turbulent
intensity normalized by uq velocity,
p
hu2
1i/uq. both plots are made with the same gray levels and
some (but not significant) decrease of this quantity is visually evident from these results. another
interesting observation from figure 6.28 is that the turbulent intensity field of the surfactant solution
seems to be ”smoothed out”, comparable to the water flow result. this result is in good agreement
with the experimental researches of van Doorn et al. (1999), Warholic et al. (2001) that observed a
significant damping of the the small scale, large amplitude velocity fluctuations using piv measure-
ments in a grid turbulence (van Doorn et al., 1999) and in a channel flow (Warholic et al., 2001) for
different surfactant and polymer concentrations.
almost all reported results with the drag reducing flows have shown that the effect of additives is
the suppression of reynolds stress in streamwise - wall-normal (x − y) plane, hu1u2i, in other words,
the one-point correlation between the streamwise and wall-normal components of the velocity fluc-
tuations. here we present the experimental results from the x−z plane (x1, x3) and, the comparison
between the water and drag reducing results of reynolds stress component in the investigated plane,
hu1u3i is shown in figure B.3. the absolute quantities (i.e., gray levels) are less important than the
fact that both sides of the figure use the same gray level, and the comparison is easily done visually.
it is clear that the reynolds stress (i.e., correlation between two velocity components) is significantly
reduced by the surfactant solution. in order to achieve more quantitative expression of the reduction,
figure B.4 presents the spanwise variation of the hu1u3i correlation averaged over the streamwise

direction, i.e.
x=xb
P
x=xa
hu1u3i, where xa, xb are the streamwise coordinates of the investigated field of
view. it is obvious that this quantity is considerably suppressed in the drag reducing case.
figure B.5 presents another effect of the surfactant solution, i.e., suppressing of the turbulent
kinetic energy production −2hu1u3is13. this reduction is even more notable than the reynolds
stresses, presented in figure B.3. the result implies that in the same regions where the surfactant
causes the decorrelation between the velocity fluctuations, it also reduces the strains in the fluid.
similar results is presented by Wei and Willmarth (1992) based on ldv measurements in a channel flow
with polymer additives. they found a dramatic reduction of the turbulent kinetic energy production
term in the x − y plane at a wall region of 40 ≤ y+
≤ 200.
summary
the drag reducing effect of the polymer additives and surfactants was investigated extensively, how-
ever, not much understanding of the underlying physical processes exists due to the complexity of
the turbulent flows and unclear interaction between the solvent and additive solution.
in the present experimental research particle image velocimetry, non-intrusive measurement sys-
tem has been applied to investigate the turbulent flow in the streamwise-spanwise plane of a flume.
the plane position at y+
≈ 80 was chosen as the most energetically important flow region, the
buffer zone, where about 80% of the energy production is done. manifestations of the surfactant
solution influence on the turbulent flow and its structure are clearly presented in the results. the
bio-degradable surfactants, from the alkyl polyglycosides family, was used as a highly efficient drag
reducing solution, however the mechanism of their effect is not known yet. the presented results
enhance the explanation of the mechanisms presented in the literature (e.g., Tsinober, 2001) and em-
phasized mostly by the decorrelation effect between the streamwise and spanwise velocity fluctuation
components and strong reduction of the kinetic energy production terms. further investigation of the
turbulent flow will focus on other planes (x−y, y −z) and stereoscopic piv measurements in order to
explain the effect of the surfactant on other reynolds stress components and production/dissipation
terms.

0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
−20 −10 0 10 20 30
−40
−30
−20
−10
0
10
20
30
Z [mm]
X
[mm]
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
−20 −10 0 10 20 30
−40
−30
−20
−10
0
10
20
30
Z [mm]
X
[mm]
Figure B.2: ensemble average of the turbulent intensity
p
hu2
1i/uq for the water (top) and
surfactant solution (bottom).

−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
−20 −10 0 10 20 30
−40
−30
−20
−10
0
10
20
30
Z [mm]
X
[mm]
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
−20 −10 0 10 20 30
−40
−30
−20
−10
0
10
20
30
Z [mm]
X
[mm]
Figure B.3: ensemble average of the one-point correlation between streamwise and spanwise
velocity fluctuations hu1u3i for the water (top) and surfactant solution (bottom).

−25 −20 −15 −10 −5 0 5 10 15 20 25
−4
−3
−2
−1
0
1
2
3
4
5
x 10
−4
Z [mm]
Reynolds
stress
〈u
1
u
3
〉
Water
Agrimul 20 ppm
Figure B.4: streamwise average of the hu1u3i correlation for the water (solid line) and surfactant
solution (star-marked line).

Figure B.5: ensemble average of the turbulent kinetic energy production term hu1u3is13 for the
water (top) and surfactant solution (bottom).

Appendix C
Principles of PIV measurement
technique
Particle image velocimetry (PIV) is a technique developed to solve an age-old problem: how to in-
stantaneously measure complex flows, over areas, without interfering with the flows themselves. PIV
is an instantaneous, non-intrusive full field flow measuring technique1
. It combines flow visualization
with quantitative measurements of instantaneous flow velocities over an extended area, providing
fluid dynamics.
The flow to be measured is seeded with suitable particles that are illuminated by a light sheet, as
shown in figure C.1. The particles must accurately follow the flow whilst scattering sufficient light
to be detectable. As the light is scattered, the motion of the gas or liquid is revealed and captured
from the side by a camera. The light sheet is pulsed and the camera captures two or more successive
light pulses in one (double-exposed) or two separate (single-exposed) images.
Figure C.1: The principle scheme of the PIV measurement technique.
The resulting images provide a displacement record of the particles within the measurement
plane which is then analyzed and scaled to velocity. The velocity information produced in this way
1The measurement is instantaneous when the time duration of the measurement is considered small (1/100)
compared to time taken for the flow to change significantly.
137

APPENDIX C. PRINCIPLES OF PIV MEASUREMENT TECHNIQUE 138
Figure C.2: The measurement system devices and their general arrangement.
tends towards the instantaneous velocity distribution as the pulse separation tends to zero. As long
as the pulse separation and camera exposure time are smaller than the smallest time scale of interest
in the flow, the measured velocities will provide a useful representation of the instantaneous velocity
field. Typically, PIV images are analyzed over a grid of local interrogation spots. The size of the
interrogation region is selected so it is large enough to include a sufficient number of particle image
pairs for an accurate measure of local displacement, but small enough so there is little variation in
velocity across the interrogation spot. A synchronizer is used to synchronize the light pulses with
the image acquisition system and to allow accurate determination of the pulse separation. The
computer, running the acquisition and analysis software, controls the acquisition system and the
synchronizer. The whole system arrangement is presented in figure C.2.

Appendix D
Principles of Stereo PIV
The SPIV is an upgrade version of the 2C2D PIV method, which is based on the principles of the
stereo photogrammetry. When the same object is viewed from two (or more) directions, it is possible
to extract its characteristics at third dimension. Following section brings the detailed description of
the particle imaging geometry reconstruction.
D.1 Particle imaging geometry reconstruction
As a first step in reconstruction process one should define the basic equations of the imaging pro-
cedure used to calculate particle displacement in planar field. The schematic view of the particle
imaging from the light sheet onto the image plane is introduced in figure D.1. For these basic cal-
culations, one may assume ideal conditions for the imaging. We might define the certain ith
particle
displacement as:
Di =
q
D2
x + D2
y + D2
z =
q
(X0
i − Xi)
2
+ (Y 0
i − Yi)
2
+ (Z0
i − Zi)
2
(D.1)
where {Xi, Yi, Zi}, {X0
i, Y 0
i , Z0
i} are initial and final particle positions, and Dx, Dy, Dz are dis-
placement projections in the planar field in three orthogonal directions, respectively. Then, the
recorded on the image plane displacement is defined as:
di =
q
(x0
i − xi)
2
+ (y0
i − yi)
2
,
139

APPENDIX D. PRINCIPLES OF STEREO PIV 140
where {xi, yi}, {x0
i, y0
i} are images of the particle initial and final positions, respectively. The initial
and final positions of the particle relates to the first (at the time t) and the second (at the time
t + ∆t) laser pulses each one.
Figure D.1: Scheme of the particle displacement imaging process
The displacement di on the image plane is calculated according to the following relations:
x0
i − xi = −M

Dx + Dz
x0
i
s

, (D.2)
y0
i − yi = −M

Dy + Dz
y0
i
s

. (D.3)
where M is the magnification coefficient, and s is the image-to-lens plane distance, as shown in
figure D.1. It is obvious that the PIV technique is very sensitive to the changes in Z direction, and
only when the laser sheet is relatively thin, the terms with Dz are small in comparison to the rest of
the equation terms, and could be neglected. However, so called ’out-of-plane’ (caused by the particle
motion in z− direction) component of the particle (i.e., fluid) velocity is one of the main error sources
in the PIV measurements. Once, the z− directional velocity is no more negligible, because the laser
sheet is not thin enough, or the third component is a desirable value, the stereoscopic imaging is the
legitimate solution.
The manipulation in an ideal case is extremely simple: the object plane is viewed by two cameras,
each one records the displacement of the particle from an angled point of view. Then the velocity
components in x and y directions by the first camera (denoted by subscript 1) are given by
U1 = −
x0
i − xi
M∆t
V1 = −
y0
i − yi
M∆t
(D.4)
In the similar manner, the velocity components at the left camera image plane are calculated
with respect to its position. In order to proceed with the velocity vector extraction, one could define
the angles α and β between the imaging axis (i.e., z axis) and the ray from the particle through the
lens optical center to the image plane, in xz and yz planes, respectively:
tan(α) = −
x0
i
s
, tan(β) = −
y0
i
s
(D.5)

By using the above definitions and relations all three components of the velocity vector could be
derived as follows (for α, β 0):
U =
U1 tan(α2) + U2 tan(α1)
tan(α2) + tan(α1)
, (D.6)
V =
V1 tan(β2) + V2 tan(β1)
tan(β2) + tan(β1)
, (D.7)
W =
U2 − U1
tan(α2) + tan(α1)
=
V2 − U1
tan(β2) + tan(β1)
(D.8)
The received formulae are general and would apply to any imaging configuration. Through usual
PIV process, the displacements from the image plane coordinate system are converted to the true
displacements in the global coordinate system, taking into account all the imaging factors. The
most significant factor is a magnification and it changes locally. Therefore, the mapping function
between the image and object planes has to be determined. The perfect solution that works for the
ideal imaging case is the geometric back-projection, which is based on geometric optics. However,
all imaging system parameters, like the lens focal length f, the position of the lens plane (which
is difficult to determine), the nominal magnification number M0 (along the principal imaging axis)
and the angles between the various planes θ, φ (see figure D.3), have to be known with high
accuracy. Unfortunately, all imaging parameters are very sensitive to small deviations of the system
configuration and calibration. Besides these difficulties, various non-linearities such as distortions
and aberrations, inherent for the real imaging system, are not taken into account. More practical and
robust approach is to ’connect’ the physical CCD-chip plane (i.e., the image plane) to the laser sheet
plane (i.e., object plane) by experimentally derived ’connection function’. There are several types of
’connection functions’ presented in the literature, like a ratios of second-order polynomials, proposed
by Willert Willert (1997), a cubic or quadratic polynomial from Soloff Soloff et al. (1997), or a bicubic
spline that has been presented by Lawson Lawson and Wu (1997), among many others. Besides
different type of the function, there also different algorithms used to calculate the coordinates.
In order to choose one of such algorithms and or functional type the extensive on-site calibration
procedure has to be done. Thus one can utilize the main advantages of these techniques that sensitive
imaging parameters not need to be determined, and in addition, distortions and aberrations are
accounted by the higher order terms of the function.

D.2 Different SPIV configurations
Generally, the PIV community has divided systems into two main categories of SPIV systems: (i)
translational and (ii) angular systems.
In the translation method the imaging systems are aligned such way that their optical axis are
perpendicular to the light sheet (similar to 2D PIV system) but view the interesting section of the
particle field far off-axis (figure D.2). In such case the images are displaced from the usual imaging
region on the camera image plane and, therefore, back planes of the cameras must be translated. The
main drawback of this stereoscopic configuration is large abbreviations due to the off-axis position,
that affects the images. In addition, when large focal length imaging lenses are used, their restricted
angular aperture limits the distance between the lenses.
Figure D.2: Schematic view of the translation SPIV system configuration with shifted imaging and
optical axis.
In the angular displacement method the imaging systems (CCD) point at the illuminated particle
field such that their optical axes (imaging axes) form the required stereo angles with the light sheet,
as the example that is presented in figure D.3.
Figure D.3: Schematic view of the stereoscopic PIV imaging system
Clearly, that the oblique orientation of the object plane with respect to the imaging plane,
results in a tilted image plane so that a normal camera will face focusing problems. The simplest
solution of such problem is, therefore, to keep the angle so small that the resulting defocusing is
still acceptable. Otherwise, in order to acquire focused images, the lens and the image plane of the
camera must be tilted one in respect to another, such they intersect in a common line with the
object plane. This is Schiempflug condition, that presented in figure D.3 and it is well established
and even implemented in commercial systems condition. The focused images, however, inherently
have a geometrical distortion. In essence the magnification factor is no longer constant across the
field of view and requires an additional means of calibration.

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19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . euqinhcet tnardauq u − v 2.5.1
20 . . . . . . . . . . . . . . . . . . . . . . . euqinhcet egareva-emit-lavretni-elbaira 2.5.2
20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . )( gnigarev ecap lavretn elbaira 2.5.3
a

b mipipr okez
21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . noitacifitnedi desab yticitro 2.5.4
22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gninoitidno :noitcete 2.5.5
23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . egareva elbmesn :noitcud 2.5.6
23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tnemecnahn langi :tnemngilae 2.5.7
23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . segatnavd 2.5.8
23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seuqinhcet noitingocer nretta 2.5.9
24 . . . . . . . . . . seuqinhcet gnilpmas lanoitidnoc tnereffid neewteb nosirapmo 2.5.10
25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . krow tnece 2.5.11
26 ihnzn rwx 3
26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeiqiqa zexcbd 3.1
27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeivplxew 3.1.1
28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qclepiix ivn‘n zrvenn dnixf 3.1.2
28 . . . . . . . . . . . . . . . . . . . . . . . . gnihcterts xetrov dna sserts sdlonye 3.1.3
29 . . . . . . . . . . . . . . . . . . . . . . . . zihpleaxehe zrvennd dnixfd ly zihpiw dibxp‘ 3.2
30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zexidndd zexfbp xefphe zeileaxr 3.3
31 dfilp‘d zyib 4
31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zihpleaxeh dnixf ly wexit 4.1
33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2
34 ieqipd jxrn 5
34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zizyz 5.1
35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SPIV zkxrn 5.2
36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dx‘dd 5.2.1
37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . drixf 5.2.2
37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zenlvnd leike zepenz zyikx 5.2.3
38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dfilp‘e oexkpiq 5.2.4
39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ieqipd i‘pz 5.3
41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -d zhiyXPIV 5.4
41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘ean 5.4.1

c mipipr okez
42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zehite‘ zenkq 5.5
42 . . . . . . . . . . . . . . . . . . . . . . . . . dx‘dd znvera dhilyl zihte‘ zkxrn 5.5.1
44 . . . . . . . . . . . . . . . . . . . . . mixeyind oia wgxna dhilyl zihte‘ zkxrn 5.5.2
45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XPIV -d ly leik 5.5.3
45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dpenzd ceairl mhixebl‘ 5.6
45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dpenzd ly micwn ceair 5.6.1
47 . . . . . . . . . . . . . . . . . . . . . . . . . . . sucof fo enalp eht ni segami elcitra 5.6.2
49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sucofed-d ixeyin ipy oia dcxtd 5.6.3
53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HFIR-e PIV ly aleyn ieqip 5.7
55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeinxz zepenz ly mcwen ceair 5.7.1
59 dfilp‘e ze‘vez 6
59 . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 − x2 xeyin zihpleaxeh dnixf ly dfilp‘ 6.1
59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zexind iqexite zeiexidn dcy 6.1.1
67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . POD micen ly zix‘ipil divpianew 6.2
68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeileaxr/zexidn ly wexit 6.2.1
68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zexidnd dcy ly lilq zlert 6.2.2
68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oey‘xd cende rvennd dcy 6.2.3
72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ilpebehxe‘d wexitd ly dxhniq 6.2.4
74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeivnr zeivwpet ly dxiga 6.2.5
76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω3 zeileaxrd aikx 6.2.6
77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ze‘vezd ly zixhnxt dxiwg 6.2.7
78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oeic - miihpxdewd mipand 6.2.8
79 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 − x3-d xeyina ze‘vez 6.2.9
80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x2 − x3-d xeyina ze‘vez 6.2.10
81 . . . . . . . . . . . . . . . icnin zlzd xefygd ici lr miihpxdew mipan ly oeit‘ 6.2.11
81 . . . . . . . . . . . . . . . . . . . . . . . . XPIV -d zkxrn mr zecicn ly zepkqne ze‘vez 6.3
83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4
83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5
83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6

d mipipr okez
91 zepwqne mekiq 7
91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1
91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2
92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . skramer gnidulcno 7.3
94 PIV zcicna dribyd jexriy ‘
94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rvenn jxr ly jexry ‘.1
95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . owzd ziihq ly jexry ‘.2
96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘.3
98 divcile a
98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dwitqd zwica a.1
99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dpkez zwica a.2
101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PIV-e LDV zekxrn oia d‘eeydd zecicn a.3
101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xeyinl avipd aikxd ly divficile a.4
103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dwicae aeyig ,zexidnd zexfbp a.5
105 zeileaxrd ayeige zexfbp b
105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . semehcs noitaitnereffid dradnat b.0.1
106 . . . . . . . . . . . . . . . . . . . . . . . . . . . semehcs noitaitnereffid evitanretl b.0.2
108 . . . . . . . . . . . . . . . . noitamitse laitnereffid ni srorre dna seitniatrecn b.0.3
110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . noitatnemelpm b.1
110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . esac tset eh b.1.1
110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . noitamitse rorre laciremu b.1.2
114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . snoisulcno b.1.3
116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tse te gnignipm - xidnepp b.2
118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . serudecorp r
balta - xidnepp b.3
124 -a dpenzd ceair zepexwr ‘
124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oefeg‘ ‘.1
124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dnxbehqid zelert ‘.1.1
125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zexfbpd mr zelert ‘.1.2

e mipipr okez
125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeibeletxen zelert ‘.2
127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . divhpnbq ‘.3
129 ghy ilirt mixneg a
137 PIV-d zepexwr b
139 Stereo PIV -d zepexwr c
139 . . . . . . . . . . . . . . . . . . . . . . . . . . . noitcurtsnocer yrtemoeg gnigami elcitra c.1
142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . snoitarugifnoc tnereffi c.2

mixei‘ zniyx
35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ieqipd zizyz ly ihnikq xe‘z 5.1
36 . . . . . . . . . . . . . . .(oinin)lr hane (l‘nyn) inciw han :mihan ipya dx‘dd znikq 5.2
39 . . . . . . . . . . . . . . . . . . . . . x1 − x2 xeyina ieqip xear SPIV -d zkxrn ly dnikq 5.3
42 . . . . . . . . . . . . . . . . . . . . .yarra lacitpo eht fo epyt tsrif eht fo weiv citamehc 5.6
44 . . . . . . . . . . . . . . . . . . . . . . . . . .ipyd beqdn zihte‘d zkxrnd ly iznikq xe‘z 5.7
46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . egami enalp eerht lanigir 5.8
47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . egami enalp eerht decnahn 5.9
:mhtirogla gniworg noiger eht ni desu elpicnirp noitcurtsnocer eht fo weiv citamehc 5.10
deifitnedi eht rof si enil )− · −( ,langis lanoisnemid eno eht swohs enil dehsa )- -(
48 . . . . . . . . . .tcejbo detcurtsnocer eht stneserp enil )− • −( ,dna slexip detarutas
49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .enalp sucof eht ni selcitrap htiw egam 5.11
50 . . . . . . . . . . . . . . . . . . . . . . . .)thgir( egami senalp sucofed dna )tfel( lanigir 5.12
51 . . . . . .)thgir( egami level yarg a sa pam tneidarg dna )tfel( egami senalp sucofe 5.13
51 . . . . . . . . . . . . . . . . . )thgir( pam tneidarg decnahne dna )tfel( egami tneidar 5.14
52 . .)thgir( egami yranib a ni stcejbo deifitnedi eht dna )tfel( egami selcitrap sucofe 5.15
53 .)thgir( evitavired sti dna )tfel( egami yranib eht fo )yrtemolunarg( noitubirtsid ezi 5.16
54 . . . . . . . . . . . . . . . . . . . . . . .HFIR -e PIV z‘ zalynd dhiyd ly iznikq xe‘z 5.17
56 . . . . . . . . . . . . . . . . . . . . . .mnegnd ghynd ly dxehxtnhd dcy ly IR-d zpenz 5.18
56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .dxehxtnhd dcy ly dkezg dpenz 5.19
57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dxehxtnhd dcy ly zllkeyn dpenz 5.20
f

g mixei‘ zniyx
58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . miizin‘d mikxrl leik mr dxehxtnh dcy 5.21
60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .u1,2 zcpezd zexidnd dcy ly dnbec 6.1
61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ũ2 -e ũ1 ly irbx zexidn qexit 6.2
62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .zrvennd zexidnd dcy 6.3
63 . . . . . . . . . . . . . . . . . . . inihxbeld wegd ly ewe xiw zecigia rvenn zexidn qexit 6.4
63 . . . . . . . . . . . . . . . . . . . . . . . u2 dna u1 zexidnd iaikx ly joint PDF zdivwpet 6.5
64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hu1u2i Reynolds ivn‘n dcy 6.6
64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dnixfd oeeka zihpiw dibxp‘ 6.7
65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiwl avip oeeka zihpiw dibxp‘ 6.8
65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −2hu1u2iS12 zihpiw dibxp‘ zxivi 6.9
66 . . . . . . . . . . . . . . .xiwl zavipd dhpicxe‘ew cbpk −2hu1u2iS12 zihpiw dibxp‘ zxivi 6.10
66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . mixeairde zeileaxrd ly mirvennd zecyd 6.11
67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .zirbxd zeileaxrd dcy 6.12
69 . . . . . . . . . . . . . . . . . . . . .)oini( zeileaxrd lye )l‘ny( zexidnd ly oey‘xd cend 6.13
70 . . . . . . . . . . . . . . . . . . . . )oini) zeileaxrd cene )l‘ny) zexidnd ly ipyd -d cen 6.14
71 .sdleif yticitrov )thgir ,ω3( gnitautculf dna )tfel ,ω̃3( suoenatnatsni eht fo sedom 6.15
73 . . . . . . . . . . . . . . ilpbehxe‘d wexitd ly zenverd qexita miixhniq micen ly iedif 6.16
74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17
)b ,3 )a ,tnenopmoc ωz yticitrov gnitautculf eht fo sedom eht fo noitanibmoc raeni 6.18
75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ylevitcepser ,sedom 051 )d dna ,01 )c ,5
76 . . . . . . . . . . . . . . . . . . . .ω3 zeileaxrd aikx ly -d icen 3 ly zix‘ipil divpianew 6.19
77 . . . . . . . . . . . . . . . . . . . . .S12 zxeaird avw ly -d icen 3 ly zix‘ipil divpianew 6.20
micen .00072 = Reh4 )b ,Reh = 24000 )a :ieqipd i‘pz xear zeileaxrd aikx ly ly mice 6.21
84 . . . . . .00045 = Reh4 )d ,Reh = 45000 )c )a :ieqipd i‘pz xear zeileaxrd aikx ly ly
85 . . . . . .zexidnd ly mixehwe htn mr cgi u1 zexidnd zeiv‘ehwelt ly mixehpew ztn 6.22
86 . . . . . . . . . . . . . . . . . . .∂(u1u3)
∂x3
zxfbpd ly dcyd rwx lr irbxd cpezd dnixfd dcy 6.23
87 . . . . . . . . . . . . .mipey mixeyin dyely xear xiwl avipd zeileaxrd aikx ly -d micen 6.24
87 . . . . . . . . . . . . . . . . . . . . . . . . x2 − x3 xeyina ω1 zeileaxrd aikx ly -d micen 6.25
88 . . . . . . . . . . . . . . . miilpebehxe‘ mixeyin dyelya zelhdk micend ly ihnikq han 6.26

h mixei‘ zniyx
89 . . . . . . . . . . . . . . .PIV -e XPIV ir cecnd ,dnixfd oeeika zrvennd zexidnd qexit 6.27
-erusaem 2 dna eht morf senalp rof u3/U1 dna u1/U1 seitisnetni tnelubrut evitale 6.28
90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .stnem
90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .η̄ fo noitubirtsi 6.29
100 . . . . . . . . . . . . . . . . . . . . . . . . InsightTM
-d zpkez mr dfilp‘e ziihpiq zpenz a.1
100 . . . . . . . . . . . . . . . . .URAPIV -e InsightTM
zepkez mr aeyign ze‘vezd z‘eeyd a.2
102 . . . . . . . .noitarugifnoc wolf dna smetsys tnemerusaem dna eht fo weiv citamehc a.3
103 . . . . . . . . . . . . . . . . . . . . .stluser tnemerusaem susrev eht fo stluser yticole a.4
104 . . . . . . . . . . .snoitarugifnoc )•( y − z ni dna )+( x − y ni derusaem eliforp yticole a.5
111 . . . . . . . . . . . . . . . . . . . . . . . . .yticitrov sti dna dleif wolf xetrov neer-rolya b.1
%5.7 )c( dna %5 )b( ,%2 )a( rof rebmun snur noitalumis fo noitcnuf a sa rorre evitale b.2
112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .level esion yticolev
0001 )c( dna ,005 )b( ,001 )a( rof level esion evitidda fo noitcnuf a sa rorre evitale b.3
113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .snur noitalumis
ytiunitnoc eht fo rorre naem eht( noitamitse niarts enalp-fo-tuo eht fo eulav nae b.4
stsisnoc tolp eh .level esion evitidda %5 dna snur noitalumis 005 eht rof )noitauqe
- ’serauq tsae’ ,thgir reppu - ’nosdrahci’ ,tfel reppu - ’retne’ eht rof stluser eht fo
114 . . . . . . .renroc thgir rewol eht ta emehcs noitaluclac ’noitalucri’ dna ,tfel rewol
egareva eht fo yticitrov eht dna sdleif yticitrov fo egareva eht neewteb ecnereffi b.5
116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .dleif yticolev
116 . . . . . . . . . . . . . . . . . . . . . . . .wolf tej gnignipmi eht fo dleif yticitrov egarev b.6
130 . . )mottob( noitarugifnoc vip dna )pot( putes latnemirepxe eht fo gniward citamehcs a.1
-rus dna )pot( retaw eht rof
p
hu2
1i/uq ytisnetni tnelubrut eht fo egareva elbmesne a.2
133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .)mottob( noitulos tnatcaf
-ev esiwnaps dna esiwmaerts neewteb noitalerroc tniop-eno eht fo egareva elbmesne a.3
134 . . .)mottob( noitulos tnatcafrus dna )pot( retaw eht rof hu1u3i snoitautculf yticol
-cafrus dna )enil dilos( retaw eht rof noitalerroc hu1u3i eht fo egareva esiwmaerts a.4
135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .)enil dekram-rats( noitulos tnat

i mixei‘ zniyx
eht rof hu1u3is13 mret noitcudorp ygrene citenik tnelubrut eht fo egareva elbmesne a.5
136 . . . . . . . . . . . . . . . . . . . . . . . . . .)mottob( noitulos tnatcafrus dna )pot( retaw
137 . . . . . . . . . . . . . . . . . . . . . . .euqinhcet tnemerusaem eht fo emehcs elpicnirp eh b.1
138 . . . . . . . . . . . . . .tnemegnarra lareneg rieht dna secived metsys tnemerusaem eh b.2
140 . . . . . . . . . . . . . . . . . . . . . ssecorp gnigami tnemecalpsid elcitrap eht fo emehc c.1
dna gnigami detfihs htiw noitarugifnoc metsys noitalsnart eht fo weiv citamehc c.2
142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .sixa lacitpo
142 . . . . . . . . . . . . . . . . . . . . . . metsys gnigami cipocsoerets eht fo weiv citamehc c.3

ze‘lah zniyx
41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .miieqipd i‘pze mixhnxt 5.1
99 . . . . . . . .dwitq cn ir dcecne PIV -d ze‘vez jezn zkxreynd dwitqd oia d‘eeyd a.1
107 . . . . . . . . . slavretni mrofinu ta decaps atad rof srotarepo laitnereffid redro tsri b.1
k

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