2. 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
8. 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
5.4 Stereoscopic PIV configuration scheme for the x1 − x3 plane experiment. . . . . . . . 40
5.5 Stereoscopic PIV configuration scheme for the x2 − x3 plane experiment. . . . . . . . 40
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
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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.
15. 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
16. 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
17. CHAPTER 2. LITERATURE REVIEW 5
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,
18. CHAPTER 2. LITERATURE REVIEW 6
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
19. CHAPTER 2. LITERATURE REVIEW 7
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
20. CHAPTER 2. LITERATURE REVIEW 8
(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
21. CHAPTER 2. LITERATURE REVIEW 9
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
22. CHAPTER 2. LITERATURE REVIEW 10
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).
23. CHAPTER 2. LITERATURE REVIEW 11
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.
24. CHAPTER 2. LITERATURE REVIEW 12
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
25. CHAPTER 2. LITERATURE REVIEW 13
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.
26. CHAPTER 2. LITERATURE REVIEW 14
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
27. CHAPTER 2. LITERATURE REVIEW 15
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)
28. CHAPTER 2. LITERATURE REVIEW 16
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
29. CHAPTER 2. LITERATURE REVIEW 17
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)
30. CHAPTER 2. LITERATURE REVIEW 18
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
31. CHAPTER 2. LITERATURE REVIEW 19
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.
32. CHAPTER 2. LITERATURE REVIEW 20
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)
33. CHAPTER 2. LITERATURE REVIEW 21
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,
34. CHAPTER 2. LITERATURE REVIEW 22
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
35. CHAPTER 2. LITERATURE REVIEW 23
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
36. CHAPTER 2. LITERATURE REVIEW 24
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
37. CHAPTER 2. LITERATURE REVIEW 25
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◦
.
38. 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
39. 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.
40. CHAPTER 3. MATHEMATICAL BACKGROUND 28
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
41. CHAPTER 3. MATHEMATICAL BACKGROUND 29
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
42. CHAPTER 3. MATHEMATICAL BACKGROUND 30
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)
43. 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
44. 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.
45. 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.
46. 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
47. 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).
48. CHAPTER 5. EXPERIMENTAL APPARATUS 36
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).
49. CHAPTER 5. EXPERIMENTAL APPARATUS 37
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.
50. CHAPTER 5. EXPERIMENTAL APPARATUS 38
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),
51. CHAPTER 5. EXPERIMENTAL APPARATUS 39
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).
53. CHAPTER 5. EXPERIMENTAL APPARATUS 41
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
