Mark_Baevsky_Msc_thesis_v2

TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of Engineering
The Zandman-Slaner School of Graduate Studies
Evolution of a turbulent patch in dilute
polymer solutions
A thesis submitted toward a degree of
Master of Science in Mechanical Engineering
by
Baevsky Mark
January 2015

TEL AVIV UNIVERSITY
The Iby and Aladar Fleischman Faculty of Engineering
The Zandman-Slaner School of Graduate Studies
Evolution of a turbulent patch in dilute
polymer solutions
A thesis submitted toward a degree of
Master of Science in Mechanical Engineering
by
Baevsky Mark
This research was carried out in The
School of Mechanical Engineering
This work was carried out under the supervision of
Prof. Alex Liberzon
January 2015

I would like to thank my teammates, Lilly Verso, Yosef Meller, Ron Schnapp
for their assistance during the experiments and an important exchange of ideas,
Mark Wasserman and Grigori Gulitski for the assistance with the hardware.
Also I would like to thank my parents and my friends for the support of my
scientiﬁc passion.
Special acknowledgment to my supervisor Professor Alex Liberzon for his
continuous involvement and careful guidance throughout my work.
This research was supported by a Grant from the GIF, the German-Israeli
Foundation for Scientiﬁc Research and Development.
i

Abstract
Drag reduction effect by dilute polymer solutions was discovered in 1946 by
Toms, but the basic mechanisms by which polymers modify the turbulent flow
have not been understood thoroughly, despite the progress in understanding the
drag reduction in pipes or channels. One of the main problems is relatively poor
understanding of dilute polymer solutions and inter-scale transfer of energy in tur-
bulent flows. The problem intensifies in the case of turbulent entrainment across
turbulent/non-turbulent interfaces on the boundaries of turbulent jets, wakes or
mixing layers. The polymer is sought to alter this region of flow significantly due
to the large gradients at the interface and strong interaction of multiple scales -
large scales that deflect the interface and the small scales that diffuse the vorticity
and strain. There is however no detailed experimental studies devoted to the inter-
faces and the numerical simulations that use polymer models (such as Oldroyd-B
or FENE-P ref. [1]) require a solid empirical background for comparison.
An experimental study has been performed to characterize the basic mecha-
nisms of turbulent entrainment in water - poly(ethylene oxide) solutions, along-
side the benchmark case of the fresh water. A new experimental setup was de-
veloped to create a spherical localized turbulent patch, thus isolating the polymer
effect far from the boundaries with negligible wall friction effects, as opposed to
the previously utilized 2D space-filling planar oscillating grids. The setup enables
a direct comparison of the results with the direct numerical simulations. We per-
formed a large set of particle image velocimetry (PIV) measurements. The patch
life cycle comprises of three phases: initial growth, a steady state and the decay
phase after the forcing have ceased. The direct polymer effect is in every stage,
from a reduced growth rate, to monotonically decreasing energy levels at steady
state and a reduced decay rate, with increasing polymer concentration (0 ppm is a
iii

freshwater benchmark case).
From enstrophy (ω2) ﬁelds we could deduce the position of the sharp inter-
face between the turbulent patch and its surrounding ﬂuid. We observe a smaller
patch, much smoother interface and the depletion of the length scales separation.
An algorithm for patch interface detection is proposed and successfully applied to
the PIV measurements, revealing the change in energy transfer towards and across
the interface, along with additional physical measures of the patch evolution. The
results will be used in developing and improving models of turbulent entrainment
and might be implemented in applications that require a precise control of local-
ized mixing rates.
iv

Table of Contents
List of Figures ix
List of Tables xv
1: Introduction 1
1.1 Goals and objectives . . . . . . . . . . . . . . . . . . . . . . . . 2
2: Literature review 3
2.1 Drag reduction by polymers . . . . . . . . . . . . . . . . . . . . 3
2.2 Turbulent entrainment with and without polymers . . . . . . . . . 4
2.3 Particle Image Velocimetry (PIV) . . . . . . . . . . . . . . . . . 6
3: Methods and materials 9
3.1 Experimental setup for set B . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 Polymer solution preparation . . . . . . . . . . . . . . . 11
3.1.3 Polymer solution parameters . . . . . . . . . . . . . . . . 12
3.1.4 Oscillating grid design . . . . . . . . . . . . . . . . . . . 14
3.1.5 PIV analysis . . . . . . . . . . . . . . . . . . . . . . . . 15

4: Results and discussion 17
4.1 Velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Vorticity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 TNTI detection using enstrophy threshold . . . . . . . . . . . . . 18
4.4 Patch area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 Patch equivalent radius . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Radial profiles of turbulent quantities . . . . . . . . . . . . . . . . 33
4.7 Turbulent kinetic energy of the patch . . . . . . . . . . . . . . . . 36
4.8 Entrainment rate coefficient . . . . . . . . . . . . . . . . . . . . . 41
4.9 Estimation of TNTI convoluted length . . . . . . . . . . . . . . . 44
4.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5: Summary and conclusions 49
5.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References 51
Appendix A: Experimental set A 55
A.1 Methods and materials of set A . . . . . . . . . . . . . . . . . . . 55
A.2 Patch area estimation method . . . . . . . . . . . . . . . . . . . . 58
A.3 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . 58
Appendix B: Energy input estimate 65
B.1 Energy input estimate using drag coefficient of the grid . . . . . . 65
B.1.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . 65
B.1.2 Grid velocity calculation . . . . . . . . . . . . . . . . . . 66
B.1.3 Drag coefficient measurements . . . . . . . . . . . . . . . 67
B.1.4 Estimated power input to the fluid through agitation using
CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
vi

B.1.5 Uncertainty analysis . . . . . . . . . . . . . . . . . . . . 74
B.2 Energy input estimate using PIV, electronics and Labview . . . . . 76
B.2.1 Motor power consumption . . . . . . . . . . . . . . . . . 77
Appendix C: Synchronization of the motor and time-resolved PIV 81
vii

List of Figures
2.1 Jets from fire hoses of water (closer hose) and with Polyox (polyethy-
lene oxide) WSR-301 (far hose). The drag reduction and diffusion
reduction helps fireman to reach twice the distance. From ref. [2] . 4
2.2 Turbulent mixing in jets of water (left) and ’Polyox’ solution (right).
From: Gadd [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Experimental setup used in experimental set B and the improved
spherical grid (d = 60mm). . . . . . . . . . . . . . . . . . . . . . 10
3.2 The polymer solution dispensing and mixing apparatus. . . . . . . 11
4.1 Vorticity magnitude maps at different stages of the patch evolu-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 An example of the patch recognition snapshots in water at agita-
tion frequency of 10.5 Hz at three time moments 1, 3 and 10 sec-
onds. The color-map is enstrophy, white dots fill the patch zone
and the red ones represent TNTI. . . . . . . . . . . . . . . . . . 21
4.3 (a) Polar coordinate system used in the post-processing of PIV
results. (b) Definition of the patch control volume area Apatch
bounded by the turbulent/non-turbulent interface, two boundaries
and the agitation device contour. . . . . . . . . . . . . . . . . . . 22
4.4 Evolution in time of the patch area, Apatch. Color curves are five
experiment repetitions and the thick black curves are ensemble
averages. Thin lines show the growth rate in the initial stage, s . . 24
ix

4.5 Patch area initial growth slope, s =
dApatch
dt
t=0
(left) versus agi-
tation frequency, (right) versus polymer concentration. . . . . . . 25
4.6 Ensemble averages of patch area versus time for all flow cases. . 26
4.7 Patch area shown in log-log scales. The bold black line illustrates
the power laws of A ∝ tn . . . . . . . . . . . . . . . . . . . . . . 27
4.8 Non-dimensional patch area Apatch. Where: f [hz] - agitation fre-
quency, s[mm2/sec] - patch growth slope shown in figure 4.4. . . 28
4.9 Equivalent patch radius as measured from the center of the agi-
tation device and defined in equation 4.2. Color curves are five
experiment repetitions and the bold black curves are ensemble av-
erage. Thin black lines show the growth slopes of the radius used
in entrainment coefficient ξ estimation using equation 4.4. . . . . 30
4.10 (a) Patch equivalent radius growth rate at t = 0 sec, (left) versus ag-
itation frequency, (b) versus polymer concentration. (right)Ensemble
average of equivalent patch radius (req) (see Eq. 4.2) in the log-
log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.11 Spatially averaged profiles of kinetic energy q(r) θ at three mo-
ments in time (1,3, and 10 sec) versus radial coordinate r for 10.5
Hz agitation frequency. All profiles start from the edge of the
agitation deice , r = 24 mm. . . . . . . . . . . . . . . . . . . . . 34
4.12 Spatially averaged enstrophy profiles, ω(r)2
θ at three moments
in time (1,3,10 seconds) for 10.5 Hz agitation frequency. . . . . . 35
4.13 Kinetic energy per unit length inside the control volume calcu-
lated from PIV velocity fields. The color lines are five experiment
repetitions and the bold black line is the ensemble average. (equa-
tion 4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.14 Ensemble average of kinetic energy per unit length inside the con-
trol volume calculated from PIV velocity fields (equation 4.3).
The color filled circles indicate the points on the curves used to
normalize the data in figure 4.15. . . . . . . . . . . . . . . . . . 39
x

4.15 Ensemble average of kinetic energy per unit length inside the con-
trol volume (CV) calculated from PIV velocity fields and normal-
ized by the steady state energy values from figure 4.14. . . . . . . 39
4.16 Values of total kinetic energy per unit length inside the control
volume (ECV − Et=0
CV ) taken from figure 4.14 and used in the nor-
malization of figure 4.15, versus squared agitation frequency ( f 2).
The thin black lines illustrate the proportionality of the values
with increasing f 2. . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.17 Entrainment coefficient ξ values between 2 and 5 seconds through
the run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.18 (a) Estimated entrainment coefficients versus the polymer concen-
tration for different agitation frequencies. (b) Estimated entrain-
ment coefficients versus agitation frequency for different poly-
mer concentrations. Calculated using equation 4.4 and averaged
through the values in figure 4.17. . . . . . . . . . . . . . . . . . 42
4.19 The entrainment rate coefficient reduction for polymer over the
water case versus grid mesh Reynolds number ReM and longest
polymer relaxation time τ calculated using eq. 3.1 in chapter 3 . . 43
4.20 Ratio of TNTI length to the arc acquired from req. . . . . . . . . 45
A.1 (left) Preliminary experimental set-up drawing. (right) drawing of
the planar grid used in set A (dimensions in millimeters). . . . . . 57
A.2 Simulink model built for image analysis of vorticity fields. . . . . 58
A.3 An example of image processing steps for the vorticity patch area
estimate: vorticity magnitude, blob analysis result, mask of the
blob over the gray scale map. . . . . . . . . . . . . . . . . . . . . 58
A.4 Vorticity snapshots for different grid frequencies for water sets
and polymer solutions, at arbitrary time t = 5 sec. . . . . . . . . . 60
A.5 Vorticity snapshots for different grid frequencies for water sets
and polymer solutions, at arbitrary time t = 5 sec. . . . . . . . . . 61
xi

A.6 Velocity magnitude from PIV realizations with upper confidence
level of 95% per field. 20 wppm experiment set (left) and 50
wppm experiment set (right). . . . . . . . . . . . . . . . . . . . 62
A.7 Patch equivalent radius calculated from the patch area for differ-
ent agitation frequencies and polymer concentrations. 20 wppm
experiment set (upper plot) and 50 wppm experiment set (lower
plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
B.1 (left) Drag measurement experimental set-up. (right) sample photo
frame of the descending spherical grid. . . . . . . . . . . . . . . 66
B.2 An example of the grid vertical position x as a function of time.
The last position points are discarded as grid exits the field of
view. The linear phase (terminal velocity) of the descend is em-
phasized using thick line. . . . . . . . . . . . . . . . . . . . . . 67
B.3 Agitation device drag coefficient measurements results. . . . . . . 69
B.4 Drag coefficient CD (grid) versus age of the polymer solution in
days since the preparation of the solution. The horizontal lines
represent the CD of water and water+glycerin solution. . . . . . . 70
B.5 Estimated power input into the flow by the spherical grid drag in
harmonic oscillations. . . . . . . . . . . . . . . . . . . . . . . . 72
B.6 Estimated RMS power into the flow from the drag forces. (left)
versus polymer concentration, (right) versus agitation frequency. . 73
B.7 Steady velocity drag power input into the flow. . . . . . . . . . . 74
B.8 The current measurement and motor actuation electrical circuit. . . 78
B.9 Power measurements comparison between PIV (left axis) and LAB-
VIEW (right axis). . . . . . . . . . . . . . . . . . . . . . . . . . 80
C.1 Hardware setup used in INSIGHT software (top), the laser fre-
quency at maximum value of 20 Hz (bottom) . . . . . . . . . . . . 82
C.2 Hardware setup used in Insight software the laser timing setup
with the synchronization signal in blue, which can be picked up at
the E/F output of the synchronizer. . . . . . . . . . . . . . . . . . 83
xii

C.3 The camera exposure signal created by the Arduino and synchro-
nized with the Insight sync signal. (Sync signal in blue, exposure
signal in red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
C.4 The ARDUINO motor trigger and camera synchronization circuit. . 84
C.5 STREAMPIX settings used to trigger the camera through the sync
in input on the camera: (left) the Optronis Control Tool is loaded
through STREAMPIX external modules window; (right) the trig-
ger settings inside the Optronis Control Tool window. . . . . . . 85
C.6 The command table example used in the motor motion programming 87
C.7 The settings used in the motor control panel. The X14.15 Enable
Manual Override X in IO Panel is to be removed to allow an ex-
ternal trigger on this line from the ARDUINO. . . . . . . . . . . . 88
xiii

List of Tables
3.1 Estimated longest relaxation times for the two polymer types used
in our experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 Viscosity of dilute polymer solutions. . . . . . . . . . . . . . . . 44
B.1 Viscosity and density of liquid solutions used in experiment set A. 70
B.2 Estimate of the power input to the ﬂow based on drag coefﬁcient
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
xv

Nomenclature
βp Entrainment constant in polymer.
βw Entrainment constant in water.
U Two-dimensional flow velocity vector as aquired from PIV analysis.
δ Entrainment rate coefficient reduction ratio.
∆l Elementary length unit.
ηs Solvent dynamic viscosity.
∀ Volume.
ν Kinematic viscosity.
ω Out of plane vorticity component.
ω2 Enstrophy from the out of plane vorticity component.
ρ Fluid density
R2
0 Root-mean-square end-to-end distance of a polymer chain.
θ Polar angle.
ξ Entrainment rate coefficient.
ζ Turbulent/non-turbulent interface surface.
xvii

A Area.
A0 Agitation amplitude.
A1 Area that is used to calculate the equivalent radius of the turbulent patch.
AL Interface projected area in the direction of propagation.
Aη Surface area of the interface resolved up to the smallest ﬂow length scales.
b Monomer length.
CPEO Polymer aqueous solution concentration.
d Cross section diameter of a stretched molecule.
E Kinetic energy per unit length.
ECV Total kinetic energy per unit length inside the control volume.
I Image intensity matrix.
i, j vector component index
K Grid action parameter.
l Single monomer length.
Larc Length of the CV sector arc of radius req.
Lmax Extended polymer chain length.
LT NTI Turbulent/non-turbulent interface length.
m Mass.
Mw Polymer weight-average molecular weight.
Mmonomer One monomer molecular weight.
n Number of monomers in an average polymer chain.
xviii

NA Avogadro number.
NPEO Number of PEO polymer molecules.
P Mechanical power.
q Kinetic energy per unit mass.
r Radial coordinate from the center of the agitation device.
Rg Radius of gyration of a polymer molecule.
ReM Grid mesh Reynolds number.
s Slope of initial growth of the turbulent patch area.
s Slope of initial growth of the turbulent patch area.
Sgrid Surface area of the agitation device geometry.
t Time starting from the begining of the agitation.
th Threshold.
u Horizontal ﬂow velocity component as aquired from PIV analysis.
uη Kolmogorov turbulent velocity scale.
ua Small scales entrainment velocity.
v Vertical ﬂow velocity component as aquired from PIV analysis.
ve Mean interface entrainment velocity.
Vgrid Vertical settling velocity of the agitation device(grid) in the settling exper-
iment.
x Horizontal coordinate.
y Vertical coordinate.
xix

PEO Poly(ethylene oxide)-polymer.
PLA Poly(lactide)-plastic.
xx

1 Introduction
Drag reduction is the most widely known effect of the dilute polymers on turbu-
lent flows, discovered by Toms in 1946. Since the time of this groundbreaking
discovery, the amount of research on possible effects of polymers on all possible
turbulent flows, mainly in the boundary layers was continuously growing. For
instance, the bibliography conducted by Nadolink and Haigh [4] contained more
than 2500 entries back in 1995. Google Scholar search of “dilute polymer drag
reduction” provides more than 24000 items. There are books that review the main
aspects of the drag reduction by additives, such as Gyr and Bewersdorff [5] and
more recent reviews such as White [6], among many others.
The drag reduction is relevant in the flows with the surface boundary layers.
There are several aspects that are related to the interaction of polymers with the
viscous sub-layer that are very different from the effects observed within turbulent
regions of the boundary layers, such as buffer and outer layers. The mechanism by
which polymers affect turbulent flows far from boundaries is not yet understood.
There is a general wisdom that turbulent flows are different in the presence of
dilute polymers, even if no drag reduction occurs. We continue in this study our
investigation of the interactions of turbulent flows with dilute polymers in a bulk of
a turbulent flow without mean velocity gradients, and far from boundaries. Since
the polymers are mostly active in presence of high strain rates at small scales,
we have chosen the case of turbulent entrainment across turbulent/non-turbulent
interfaces, relevant for the turbulent jets, wakes or mixing layers, to be the focus
of this study.
The most interesting questions are related to the process of turbulent entrain-
ment related to the adjustment of the interface region to the different turbulent flow
conditions inside the turbulent region. One could expect that stronger anisotropy
1

of the turbulent flow will affect strongly the processes at the interface. It is also
possible that weaker velocity gradients at the interface are compensated by the
faster propagation of the larger scales and exhibit higher entrainment rate. One
would have to measure in great details the production and destruction of turbulent
kinetic energy at various distances from the interface, turbulent quantities such
as Reynolds stresses, vorticity and other key properties that underline the turbu-
lent flows. There is very little information available about the turbulent proper-
ties of the entrainment layer and this information is crucial for validation of the
sophisticated models (such as Oldroyd-B or FENE-P ref.[1]) used in numerical
simulations of polymers.
1.1 Goals and objectives
The major goal is to study with greater details the effect of dilute polymers on
the formation and evolution of turbulent flows with an interface across which the
entrainment of non-turbulent flow occurs. The detailed objectives are:
1. to apply a non-invasive experimental method to the turbulent flows with and
without dilute polymers, in presence of a turbulent/non-turbulent interface
and obtain spatial velocity and velocity gradients information .
2. estimate the effects of polymers on the entrainment properties such as the
entrainment rate and the geometrical characteristics of the interface, as a
function of forcing, polymer concentration and molecular weight.
3. provide a benchmark test case for the numerical simulations that attempt to
model mixing and turbulent entrainment in free shear flows.
2

2 Literature review
2.1 Drag reduction by polymers
Drag reduction effect of dilute solutions of polymers was discovered in 1946 by
Toms (and published in 1948 see [7]). Since than many experiments were done
addressing this phenomenon, called “Toms-effect”, primarily in the field of pipe
flows and wall boundary layers of submerged bodies. Theoretical models were
proposed to describe this effect in a universal way but with limited success be-
cause of the limited understanding of two key ingredients: physics of dilute poly-
mer solutions and physics of turbulent flows. The drag reduction is relevant only
in turbulent flows as was described in details by experiments of pressure drop ver-
sus flow rate performed by Virk [8]. It was shown that in laminar regime regular
Poiseuille friction law is obeyed by all polymer solutions, after transition to tur-
bulent flow the data follows Prandtl-Karman law for Newtonian fluids, and only
above some threshold of Reynolds number and above a certain wall shear stress, it
deflects from Prandtl-Karman law in the direction of lower friction. The drag re-
duction increases with increasing Reynolds numbers, but maximum possible drag
reduction that can be achieved is limited by an asymptote, called maximum drag
reduction (MDR asymptote), which has a similar to Prandtl-Karman law form.
An example of the drag reduction effect can be seen on figure 2.1 where exploit-
ing the same pumping power the jet of water from the fire hose with a polymer
additive (polyethylene oxide in this case) reaches twice the distance of the regular
water jet. The main effects were observed in the hoses themselves, yet there is
also a clearly visible effect on free jets. For instance, in figure 2.2 two submerged
water jets could be seen where on the right the polymer solution evidently shows
slower dye diffusion and strongly reduced entrainment of surrounding fluid.
3

Figure 2.1: Jets from fire hoses of water (closer hose) and with Polyox
(polyethylene oxide) WSR-301 (far hose). The drag reduction and diffu-
sion reduction helps fireman to reach twice the distance. From ref. [2]
.
Figure 2.2: Turbulent mixing in jets of water (left) and ’Polyox’ solution
(right). From: Gadd [3]
2.2 Turbulent entrainment with and without polymers
Turbulent entrainment is a process of continuous transition of the fluid from non-
turbulent to turbulent flow across a typically thin interfacial layer (turbulent/non-
4

turbulent interface, abbreviated hereinafter TNTI) [9, 10]. The process of entrain-
ment is essential in many natural and engineering flows such as boundary layers,
jets, plumes, wakes and mixing layers. The main distinction between the turbu-
lent and non-turbulent regions [11, 12], is that turbulent regions are rotational, i.e.
ω 0, where vorticity is a curl of velocity vector field ω = ×u. Non-turbulent
flows are irrotational, i.e. the vorticity is zero, ω = 0.
There are typical approaches to entrainment problem. One point of view relies
on the large scale analysis. The evolution of the TNTI at large is defined by the
large scales of the turbulent flow inside the turbulent region, as the non-turbulent
region is engulfed by the turbulent region. An alternative view emphasizes the
small scale effect of viscous diffusion of vorticity which turns the non-turbulent
fluid regions into a turbulent state. The effect of large scales versus the small
scales (nibbling of the interface vs engulfment) is a topic of an ongoing scientific
debate [13].
There are also two measurement methods for the entrainment rate (or entrain-
ment coefficient), defined using large scales or small scales approaches. For in-
stance in the turbulent jet one can measure the width of the jet as a function of
streamwise distance and describe the ratio in terms of the turbulent front propa-
gation velocity, υe relative to the mean jet velocity U. One can also measure the
local features of the interface like the shape of TNTI (ζ(x,θ,t) for an axisymmet-
ric jet) and the local entrainment velocity relative to the energy that is contained
inside the turbulent region just behind the interface. An entrainment coefficient
is typically defined as β = ve/vrms which represents the ratio of the propagation
velocity of TNTI to the turbulent kinetic energy vrms right behind TNTI . It is
important to distinguish between the entrainment velocity ua and the propagation
velocity υe. The latter is the advancement of the mean position of the interface
toward the non-turbulent region, which is determined by the energy-containing
eddies in the flow through the process of engulfment. Velocity ua is of the order
of the Kolmogorov velocity scale uη, and is closely related to the nibbling process
of the interface. The two velocities are related through the interface geometry by
the following relation[9]:
υe = ua 1+ ( ζ)2
1/2
5

And the relation between υe and uη could be written in terms of entrainment
flux, υe AL = uη Aη, where AL is the interface projected area in the direction of
propagation, and Aη is the surface area of the interface resolved up to the smallest
length scales, Liberzon et al. [14].
Turbulent entrainment with polymers has been found attractive as it allows
to study the direct effect of dilute polymers on the transition to turbulence, dif-
fusion of turbulence with practical applications in drug delivery, chemical pro-
cess control and drag reduction among others. Liberzon et al. [14] used a pla-
nar space filling oscillating grid to generate a propagating TNTI, and measured a
faster-propagating TNTI in polymer solution relative to the fresh-water case due
to higher grid action parameter (Kp > Kw, where K defined as H =
√
Kt, H -
being the average TNTI position as the function of time - t) but lower entrainment
constant βw = 0.8, βp = 0.7. Also the authors [14] found larger integral veloc-
ity and length scales in polymer solutions. The dissipation fraction of turbulent
kinetic energy by polymers was found to be 28% from total dissipation rate. In-
teresting to note that as those experiments were carried out using a space filling
grid the turbulent region had contact with the walls of the water tank, thus bound-
ary layers, although weak, could have influenced the energy balance inside the
turbulent region. Wu [15] implemented another technique where a spiral paddle
was released to swing freely through a water tank creating an almost cylindrical
turbulent patch, a cross-section of the patch was photographed and it’s area calcu-
lated as a function of time. The entrainment characteristics were not measured in
this work but authors noted the polymer effect on the production and dissipation
characteristics of free turbulence, causing less production and more dissipation.
In this study we will utilize the same approach as in Ref. [14], though in a dif-
ferent setup, applying particle image velocimetry (PIV) to quantify the turbulent
flow, entrainment and characteristics of the TNTI in dilute polymer solutions.
2.3 Particle Image Velocimetry (PIV)
Particle image velocimetry (PIV) is a method of acquiring instantaneous two di-
mensional (in plane) vector fields of velocity. This method is based on acquiring
two sequential images of small particles moving with the flow and estimating the
6

displacements in different regions of these two images, typically by means of
cross-correlation. Dividing the displacements by the time period (delay) between
the two images one obtains the two-component, two-dimensional velocity vector
field. By capturing a sequence of images (like a video) and analyzing each pair of
images we can resolve the evolution of the flow field in time [16].
In order to obtain high fidelity velocity fields from cross-correlation analysis
of images, the small seeding particles have to fulfill two conditions : particle
inertia must be small enough compared to drag forces and particle must be as
close to neutrally buoyant as possible [17, 18, 19]. Thus the particle should be
small enough yet made from material that provides the sufficiently strong intensity
of light scattered by the particle.
In order to keep the seeding particles small and in order to focus the light in to
a thin planar sheet, a high intensity laser is used for illumination of a plane through
the flow. Usually a Nd:YAG laser (λ=532nm, green light) is used, the beam is gen-
erated by Nd3+ions incorporated into YAG (yttrium-aluminum-garnet) crystals.
The excitation is performed by optical pumping with white light flash lamps. The
laser incorporates two pumping chambers, each one of these contains an Nd:YAG
rod and a flash lamp, this arrangement allows very short time delay between two
laser pulses. The amount of energy released by the laser is controlled through al-
tering the quality factor of resonant cavity, this is done by the so called Q-Switch.
In this way Q-Switch controls the timing of laser pulses and also the amount of
energy in each pulse. After exiting output beam aperture, the beam passes through
spherical diverging and cylindrical lenses to form a thin laser sheet. Recently also
Nd:YLF solid-state lasers have been used for PIV, yet the laser beam and the sheet
light quality is lower than that of Nd:YAG. The main advantage of the Nd:YLF
lasers is their high repetition rate, up to tens of kHz.
The acquired images are divided into small regions, called interrogation win-
dows. For each interrogation window (I) in the first image there is a correspond-
ing and larger window (I ) in the second image with it’s center coincident with the
center of the smaller window. The displacement within the interrogation window
is estimated as a position of the maximum peak of the two-dimensional cross-
correlation function (sub pixel displacement is evaluated by interpolation) [16]:
7

RII (x,y) =
K
i=0
L
j=0
I(i, j)I (i + x, j + y) (2.1)
where K and L describe the size of window I. In practice the cross-correlation
function is implemented using the Fast Fourier Transform (FFT) and the convo-
lution theorem which states that the cross-correlation of two functions is equiva-
lent to a complex conjugate multiplication of their Fourier transforms:RII ⇐⇒
Î ˆ⊗I
∗
where Î and Î are Fourier transforms of I and I , respectively. After
complex-conjugate multiplication and inverse Fourier transform we receive the
spatial cross-correlation which is equivalent to RII in equation 2.1.
8

3 Methods and materials
The results in this work obtained in two rounds of experiments: A) experiments
with a DC motor and a small planar grid described in detail in appendix A, and
B) experiments with a spherical grid and a linear motor described in this chap-
ter. Additional major difference between the two sets is the polymer: in set A we
used polyethylene oxide with a molecular mass of 4 × 106 and in set B we used
a longer monomer chains polyethylene oxide with a molecular mass of 8 × 106.
Consequently different concentrations of polymer solutions were needed for the
effect, 25-50 wppm in the set A and 2.5-10 wppm in the set B. This chapter de-
scribes the experimental setup and methods used in set B (the last one). Set A is
described in details in Appendix A. The principle of the experiment in both sets
is to create a turbulence patch (as much isotropic and homogeneous as possible)
in the tank filled with water or polymer dilute solution, and to quantify the flow
through the two-dimensional particle image velocimetry (PIV) method. In addi-
tion, we have measured the actual steady state drag force and the drag coefficient
of the spherical grid to quantify the energy input to the flow.
3.1 Experimental setup for set B
3.1.1 Experimental setup
The experimental setup layout can be seen on Figure 3.1. The tank of square
cross-section 300 × 300 mm is filled with water or water-polymer solution up to
300 mm level. An agitation device, the grid of 6 cm in diameter, (see section
3.1.4 for the details) is set to oscillate precisely in the middle of the water vol-
ume at a stroke of ±2.5 mm. The grid is attached to the motor shaft through
9

two stainless steel rods (all parts covered to prevent the laser light reflections).
Three frequencies are used for different sets of experiment: 6.86, 8.40, and10.53
Hz. The flow is seeded using hollow glass spheres of 10 µm average diameter
(POTTERS INDUSTRIES INC.). A nearly spherical turbulent patch is observed in
the water volume as the grid is oscillated at those frequencies. The two lasers are
synchronized in a specific way to illuminate the flow field at 40 pulses/sec. OP-
TRONIS CL4000CXP digital high speed CMOS camera is synchronized with the
laser pulses, when the exposure starts 1×10−3 sec before the laser pulse and ends
23 millisecond after the pulse. The motor is synchronized with the PIV system
to start exactly at the rising edge of the first exposure signal (no motor frequency
ramping) and is kept oscillating for 20 seconds. The PIV capture sequence is left
running for approximately 10 more seconds in order to measure also the decay
stage of the flow. The custom-made synchronization setup is described in details
in appendix C.
600[mm]
Experiment Tank
520[mm]
60[mm]
300[mm]
300[mm]
Laser
Grid
Camera
300[mm]
300[mm]
150[mm]
Linear servo motor
Direction of
OscillationsLaser sheet
Top view
Side view
Figure 3.1: Experimental setup used in experimental set B and the im-
proved spherical grid (d = 60mm).
10

The Geared DC Motor
The Dispenser Funnel
A Polymer Powder
The Stirrer Rotor
1[hz]
The Vibrational DC Motor
Figure 3.2: The polymer solution dispensing and mixing apparatus.
3.1.2 Polymer solution preparation
The polymer that was used in current experiment is E-500C poly(ethylene oxide)
with a molecular weight of Mw > 8,000,000. Polymer preparation is known to
be one of the problematic factors in repeatability of the dilute polymer solution
experiments. For that reason, an automated preparation process was developed
in which the powder dispensing and the stirring actions were electronically con-
trolled. The automatic powder dispenser machine consists of a conical funnel
with a 1 mm circular opening in the bottom, with a correct amount of polymer
needed for the specific concentration. A coin cell battery sized vibration motor is
attached to the funnel and is connected through a MOSFET transistor to an AR-
DUINO controller board programmed to activate the motor with a specific timing
for the whole preparation process. The stirrer is automatically stopped every 30
seconds and the polymer powder is deposited by the funnel automatic vibration,
as shown in figure 3.2. The primary (parent) solution was dissolved in water at
room temperature.
11

3.1.3 Polymer solution parameters
Polyethylene oxide (PEO) used in the major part of the study is the E-500C,
produced by Alkoro GmbH. According to the manufacturer, it has the average
molecular weight of approximately Mw > 8000000[g/mol]:. Its monomer
formula is C2H4O1 and molecular weight per monomer is
Mmonomer = 44[g/mol]. Number of monomers in a chain can be estimated as:
n = Mw/Mmonomer = 181818. Single monomer length, following Mark et al. [20]
is l = 3.6×10−10 m, while fully extended chain length can be estimated as
Lmax = l · n = 65.4[µm]. Fetters et al. [21] reports that the root-mean-square
(RMS) end-to-end distance R2
0 for poly(ethylene oxide) is R2
0
Mw
= 0.805,
thus R2
0 =
√
Mw ·0.805·10−10 = 253.77[nm].
Using the above information, we can estimate the important factor in our study,
the overlap concentration c∗ . In this work we can use the formula from Doi et al.
[22]:
c∗
=
3M
4πNAR3
g
where NA = 6.022·1023[1/mol] is Avogadro number. From Ref.[23] we can get
also the radius of gyration of Rg = 0.0215· M0.583 = 227.478[nm]. Thus we can
estimate the overlap concentration of PEO as:
c∗
=
3·8·106
4π ·6.022·1023 · (227.478·10−9)3
= 269.426
µgr
mL
= 0.269
gr
L
These values allow us to estimate how dilute is the concentration in our
experiment. For 10ppm E-500C in 27 liter tank (set B experiment) the
concentration is:
C
10ppm
exp =
mPEO
∀water
=
0.27[gr]
27[L]
= 0.01
gr
L
12

Hence we can assure that in this experiment, preparing a homogeneous so-
lution, we achieve with very dilute suspension with a concentration an order of
magnitude lower than the overlap concentration.
Second important parameter that we need to estimate is the probability of
interference of the long polymer molecules. There are two states of polymer
molecules relevant for this analysis, namely the free polymer chain in a coiled
state (with a known radius of gyration) and a hypothetical state of fully stretched
molecules to their maximum length. To estimate the possibility of interference
or entanglement we can calculate the effective volume that would be occupied
by the polymer molecules that is assembled from cubic volumes containing one
molecule each and of a length scale equal to the stretched end-to-end size of the
polymer molecule and compare that volume to the actual volume of solvent in
an experiment. For free-coiled chain the appropriate length scale would be the
(RMS) end-to-end distance of the chain, and for a stretched chain it is the maxi-
mum linear length of the molecule.
The number of molecules in the tank for 10ppm E-500C polymer in set B
experiment, NPEO = mPEO
Mw
· NA = 2.032 · 1016. Estimated cubic volume occu-
pied by one molecule as ∀molecule = R2
0
3
= 1.63 · 10−20[m3], we can ﬁnd
the total volume occupied by the polymer molecules ∀PEO = ∀molecule · NPEO =
3.32 · 10−4[m3], and as a result their volume fraction in the unstretched case is as
follows:
∀PEO
∀water
= 0.0123
We infer from this ratio that there is a little chance for the molecules to inter-
fere with each other in this state. If, however, hypothetically one can stretch all
the molecules to the maximum length, then the effective volume of a cube that
contains one molecule is ∀molecule = (Lmax)3
= 2.8 · 10−13[m3] and then the total
effective volume is unrealistically high, ∀PEO = ∀molecule · NPEO = 5.68·103[m3].
Obviously this is a little physical meaning in these numbers as the highly stretched
molecules are very thin cylinders with approximate diameter of d = 2.13 Å, yet
it provides a hint to the possible mechanism of action - if the chemical bond be-
tween two molecules is high enough, there is a chance that after each collision of
13

two molecules, a network is created which is increasing in volume at very high
rate as the shear continuously stretches the molecules. This can explain, for in-
stance, hypothetically the reason for the shear stress threshold in drag reduction
studies which is sharp, and the related to it hysteresis of the effect when the shear
is removed, yet the drag reduction continues. To summarize, we can infer that in
a stretched state there is a much higher possibility for the molecules to interfere
and to create the entangled networks.
Another key parameter of the polymer is the longest relaxation time of the
polymer molecules in a speciﬁc solution. As we saw above there is a good chance
for the molecules to entangle when a stretching strain is applied to them hence the
relaxation time of the molecules will depend on the concentration of the solution
as well, because the chains might come close enough to each other so that viscous
diffusion can transfer information between the chains. We will use the following
relaxation time estimate found in ref. [24](eq. 2):
τ ∼
ηsb3
kBT
n3ν c
c∗
(2−3ν)/(3ν−1)
(3.1)
Where kB = 1.38e −23 m2kg
sec2K
is the Boltzmann’s constant, T is the tempera-
ture of the solution, ν = 0.588 for a good solvent, c is the solution concentration, c∗
is the overlap concentration of the polymer (calculated in preceding paragraphs),
b is the monomer length, ηs is the dynamic viscosity of the solvent. These results
help us to understand that at shear rates or strain rates of this order of magni-
tude, the polymers can effectively react and extract or transfer the energy from the
turbulent ﬂuctuations.
3.1.4 Oscillating grid design
An agitation device (or the spherical grid) was designed by trial-and-error ap-
proach. Numerous trials were performed to create an almost spherical turbulent
patch with axisymmetry. The main objective was the homogeneous approach and
averaging of the patch properties at all angles, leading to a 1D propagation prob-
lem. The spherical shape makes TNTI recognition and comparison to numerical
simulations results much more convenient - the numerical counterpart can be any
14

Polymer WSR-301 concentration τ[msec]
20 wppm 2.15
50 wppm 2.86
Polymer E-500C concentration τ[msec]
2.5 wppm 4.51
5 wppm 5.59
10 wppm 6.93
Table 3.1: Estimated longest relaxation times for the two polymer types
used in our experiments.
other homogeneous front propagation with low curvature. The agitation device is
shown in figure 3.1. The grid was printed in 3D from PLA plastic. The agitation
device contains two cambered profiles extruded by revolution through 360 de-
grees around the vertical axis of the grid to form two hydrodynamic surfaces. By
moving the grid up and down through the fluid, these surfaces deflect the fluid in
radial direction and create almost spherical turbulent patch in water. It was found
that the grid is creating the most symmetric patch at a peak-to-peak stroke of 5
mm and at frequencies of 6÷11 Hz. The diameter of the device is 6 centimeters.
3.1.5 PIV analysis
The images (4 Mp images at 40 frames per second) were stored on hard drives
and processed using OPENPIV Taylor et al. [25]. In addition, a dynamic masking
algorithm was implemented to mask the grid and the rods. PIV images were
processed with 32×32 pixel interrogation windows and 50% overlap (the second
image search area was 64×64 pixels). Data was filtered using global filter with a
threshold of 10 times the mean value, approximately 300 pixels/∆t, followed by
a 3×3 local median filter and interpolation with the same kernel size. Additional
spatial median filter is applied to each of the vector fields with a window size of
3 × 3 grid points, in order to replace the large erroneous vectors and smooth the
data for the following analysis using PIVMAT toolbox.
15

4 Results and discussion
In this chapter the PIV data for the set B is analyzed and discussed. Starting with
the velocity and vorticity fields we proceed to the analysis of the patch evolution,
focusing on the growth rates and energy transfer rates of the patch. The central
method is the comparison of the Newtonian flow cases of water with the non New-
tonian dilute polymer solution cases with an attempt to reveal the key mechanisms
of polymer-turbulence interaction.
4.1 Velocity fields
Using PIV analysis as described in chapter 3, we obtain the two-component ve-
locity realizations at each time step, U(x,y,t) = {u(x,y,t),v(x,y,t)}. The fields
include 85 × 99 vectors with spatial resolution of 1.42 mm, covering a field of
view 123×144 mm around the spherical grid.
A temporal moving average filter (PIVMAT toolbox) with the cut-off frequency
of 1 Hz was applied to filter-out the grid oscillation frequencies:
Û(x,y,tk ) = U(x,y,tn)n=k,(k+1),(k+2),...,(k+W−1) (4.1)
where Û is the filtered velocity field sequence, and Un denotes the time mov-
ing average of the fields listed in n. As a result Û(x,y,tk ) is shorter than the
original sequence by W −1.
17

4.2 Vorticity fields
Applying numerical derivatives to the velocity fields, the out-of-plane component
of vorticity fields, ω(x,y,t) = ∂v(x,y,t)
∂x − ∂u(x,y,t)
∂y , are calculated [16]. Figure 4.1
presents some samples of the patch evolution process highlighted by the mag-
nitude of vorticity, |ω(x,y,t)| for different polymer concentrations and agitation
frequencies. From sub-figure 4.1a the evolution in time could be recognized (from
three moments in time 1,3 and 10 seconds since the start of the agitation) as the
patch grows in size, the vortices are propagating into the quiescent fluid. In sub-
figures 4.1b,4.1c and 4.1d we compare the effect of different agitation frequency
for three polymer concentrations, all at the time snapshot t = 10 sec. We note how
the polymer visco-elastic effects of 5ppm solution limit the patch spreading in
the horizontal direction and mainly redirect the flow in the vertical direction that
aligns with the direction of oscillation. In 10ppm solution flow case even larger
portion of energy is contained in two large vortices that do not propagate and re-
main at short distance from the oscillating grid. There is also a shear layer that
apparently connects the two large vortices , clearly visible in the 10ppm case. The
reader shall realize that the masked region of the oscillating grid is larger than the
grid itself by the peak-to-peak amplitude and removed during the PIV analysis.
There are strong flow-grid interactions in this region in the form of jets and wakes
entering and leaving the grid as the oscillation continues.
4.3 TNTI detection using enstrophy threshold
The study focuses on the evolution of a turbulent patch, thus we need to define
and identify the turbulent/non-turbulent interface (TNTI). The identification will
allow in addition to the analysis of TNTI itself, also to estimate the energy content
of the patch, its size and area growth in time (shown below in Sections 4.4, 4.5,
4.7) and subsequently to calculate the entrainment rate coefficient (in Section 4.8).
The turbulent region can be distinguished from the non-turbulent one using enstro-
phy (ω2) threshold technique that appears to be the most robust approach see for
instance Refs. [12, 26, 10, 14]. We use an ad-hoc developed method, similar to
those proposed in the literature, mainly due to a lack of a single, objective selec-
tion method. Although the method relies on a somewhat arbitrary choice of the
18

threshold, it was carefully validated for robustness and was found to perform well
for a given set of data with low sensitivity of the following results to the particu-
lar choice of the threshold value. The enstrophy ﬁelds ω2(x,y,t) are normalized
by the maximum enstrophy value in each frame:(ω2(t))max = max(ω2(x,y,t)),
resulting 0 ≤ ω2(x,y,t) = ω2(x,y,t)
(ω2(t))max ≤ 1. A median value of the normalized
enstrophy ﬁeld is calculated Median(t) = median( ω2(x,y,t) ), and a thresh-
old th = 20 is chosen as the one that provides the most robust turbulent zone
boundaries for all experimental runs (chosen by trial and error), to create a mask
Imask (x,y,t):



ω2(x,y,t) ≥ th · Median(t) : Imask (x,y,t) = 1
else : Imask (x,y,t) = 0
19

x [mm]
y[mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
|ω|(s−1
)
0
1
2
3
4
5
(a) water at 6.9 Hz, t=1, 3, 10 seconds, from left to right respectively.
x [mm]
y[mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
|ω|(s−1
)
0
1
2
3
4
5
(b) water 5 ppm and 10 ppm at6.9 Hz , t=10 seconds
x [mm]
y[mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
|ω|(s−1
)
0
1
2
3
4
5
(c) water 5 ppm and 10 ppm at8.4 Hz , t=10 seconds
x [mm]
y[mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
|ω|(s−1
)
0
1
2
3
4
5
(d) water 5ppm and 10 ppm at10.5 Hz , t=10 seconds
Figure 4.1: Vorticity magnitude maps at different stages of the patch
evolution.
20

x [mm]
y[mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
x [mm]
20 40 60 80 100120
20
40
60
80
100
120
140
ω2
(s−2
)
0
5
10
15
20
Figure 4.2: An example of the patch recognition snapshots in water at
agitation frequency of 10.5 Hz at three time moments 1, 3 and 10 sec-
onds. The color-map is enstrophy, white dots fill the patch zone and the
red ones represent TNTI.
This mask per field is used later in all the following analysis to identify the
turbulent zone, when zeros represent the quiescent fluid outside and neglected.
An example of the recognition algorithm result could be seen in figure 4.2, where
the patch development is automatically followed by the recognition algorithm and
it is able to provide the extent of the TNTI and the patch internal area at x-y
plane. The white dots are from the (xCV (t),yCV (t)) coordinates subset such that
Imask (xCV (t),yCV (t),t) = 1 and the red ones representing the TNTI itself are de-
fined as the position of transition from Imask = 1 to Imask = 0.
After TNTI definition, we can describe the evolution of the patch in terms of
growth of its area or equivalent distance from the source (similar to the depth of
the turbulent region, h(t) used in homogeneous grid experiments). Since the patch
created by a quasi-spherical grid, the following analysis is performed in polar co-
ordinates where the coordinate system is defined in figure 4.3. The mid-position
of the grid is the origin r = 0 with the positive angle θ measured from the hori-
zontal coordinate, x, in clockwise direction. A thick convoluted line denotes the
instantaneous TNTI position. There is a problem defining the patch in the vertical
directions (θ ≈ ±90o). In these directions, the flow moves mostly under the the
pressure field due to oscillation of the grid, rather than due to turbulent motions.
In order to isolate our analysis from these inevitable and undesired effects, we
limit our study to the control volume of the patch as defined in the following.
21

Agitation device
Theta-coordinate
r-coordinate
(a)
y=72.6[mm]
y=-72.6[mm]
x
y
x=-1.4[mm]x=-125[mm]
Artificial Boundary 2 (B2)
Artificial Boundary 1 (B1)
58 [degrees]
Control volume (CV)
TNTI
Agitation device
(b)
Figure 4.3: (a) Polar coordinate system used in the post-processing of
PIV results. (b) Deﬁnition of the patch control volume area Apatch
bounded by the turbulent/non-turbulent interface, two boundaries and
the agitation device contour.
22

4.4 Patch area
Patch area is the area within the control volume as shown in figure 4.3b, bounded
by the TNTI and the θ = ±58o . The angle is chosen so as to minimize the effects
of the vertical flow phenomena on the analysis of all sets. The area is estimated
counting all the interrogation windows within the control volume, as follows:
Apatch(t)
i
=
x(t) y(t)
A
where sets of coordinates x(t),y(t) are the points which lie inside the control
volume defined in figure 4.3a and A = 1.422 · 10−6 [m2] is the elementary area
derived from spatial resolution of the PIV realizations. The patch area evolves
with time as shown in figure 4.4, where each thin line is the result of a single run
and the thick line is the ensemble average. The thin line at the origin emphasizes
the growth rate during the initial growth period. We observe an increase of area
with time as expected in all solutions, while the water case exhibits much stronger
turbulent flow and faster growth of the patch area, as well as larger maximum area.
There are several peculiar effects such as the reduction in area in the water flow
case after 10 seconds (although the forcing stops only after 20 seconds). This is
apparently due to appearance of zones with low enstrophy inside the patch which
are the result of 3D motions and excluded from the total patch area count due to
the non-adaptive type of the threshold in our algorithm. In contrast , the polymer
solutions exhibit a steady state phase till the 20 seconds that starts at different
times for different frequencies. It may be a result of a weaker turbulent flow in
polymer and therefore weaker 3D effects. A thin black line on the plots shows the
initial slope of the area growth for all cases. The patch area is smaller for higher
concentration of the polymers as compared to the water flow case.
In figure 4.5 the values of the patch area growth rate at the initial phase of
t = 0 ÷ 5 seconds are plotted versus the agitation frequency and as a function of
the polymer concentration (the values are taken from the thin lines slopes marked
in figure 4.4). We observe that the growth rate is increased with frequency, as
expected. The dependence of the growth rate of area on concentration is less ob-
vious: for the weak forcing at 6.9 Hz the growth rate decreases monotonically
23

0 10 20
0
2000
4000
(a)
0 10 20 30
0
2000
4000
(b)
0 10 20 30
0
2000
4000
(c)
0 10 20 30
0
2000
4000
(d)
Apatch[mm2
]
0 10 20 30
0
2000
4000
(e)
0 10 20 30
0
2000
4000
(f)
0 10 20 30
0
2000
4000
(g)
t [sec]
0 10 20 30
0
2000
4000
(h)
t [sec]
0 10 20 30
0
2000
4000
(i)
t [sec]
a) water 6.9 Hz b) water 8.4 Hz c) water 10.5 Hz
d) 5ppm 6.9 Hz e) 5ppm 8.4 Hz f) 5ppm 10.5 Hz
g) 10ppm 6.9 Hz h) 10ppm 8.4 Hz i) 10ppm 10.5 Hz
Figure 4.4: Evolution in time of the patch area, Apatch. Color curves
are ﬁve experiment repetitions and the thick black curves are ensemble
averages. Thin lines show the growth rate in the initial stage, s .
24

7 8 9 10 11
100
200
300
400
500
f [Hz]
s[mm2
/sec]
0 5 10
100
200
300
400
500
CPEO [wppm]
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.5: Patch area initial growth slope, s =
dApatch
dt
t=0
(left) versus
agitation frequency, (right) versus polymer concentration.
with the concentration. However, for stronger forcing, the 5 ppm solution appears
to be more effective in reducing the growth rate than the 10 ppm. This can indi-
cate a contributions of different effects related to the energy transfer in this flow
system: from oscillating grid through jets/wakes/friction to the turbulent flow and
from turbulent kinetic energy to the non-turbulent fluid and dissipation.
Figure 4.6 summarizes the ensemble averages of the patch area (the thick
curves from Fig. 4.4) from which the initial values at t = 0 are subtracted for
the sake of comparison. The area of the patch in the water case is twice as high as
in the polymer solutions, as well as the area growth rate, dA/dt, is much stronger.
The growth rate of the patch area is only slightly increased with increasing agi-
tation frequency. It is important to note that our analysis is sensitive to the local
events, such as seen in Fig. 4.6 the beginning of the 10 ppm curve for 6.9 Hz
frequency. The initial growth rate is apparently very high, but careful visualiza-
tion of the flow maps revealed that there was some local and transient flow with
larger jet/vortex was ejected from the grid (this local transient event feature will
be visible in all of the figures describing the time profiles of different quantities).
In the next figure 4.7 the same data from figure 4.6 is shown in a log-log axes
with an attempt to deduce a power law from the time interval of monotonic area
growth. A power law in the form Apatch ∝ t0.4 is found to be valid by all frequen-
25

0 5 10 15 20 25 30
−500
0
500
1000
1500
2000
2500
3000
t [sec]
Apatch−At=0
patch[mm2
]
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.6: Ensemble averages of patch area versus time for all ﬂow
cases.
26

10
0
10
1
10
2
10
3
t [sec]
Apatch[mm2
]
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
t0.3
t0.4
Figure 4.7: Patch area shown in log-log scales. The bold black line illus-
trates the power laws of A ∝ tn
cies in the water flow case, this can strengthen the view that the patch growth is
a self-similar process, independent of different frequencies and predefined by a
single set of scales. The polymer curves show that the patch growth rate is equal
to that of water for the first 2 seconds, then it is lower for the rest of the growth
period (up to 10 seconds approximately), ∝ t0.3. It also appears to be the same rate
for 5 and 10 ppm (except a single case of the weakest forcing and highest concen-
tration), as we have seen in Fig. 4.5. It remains to be understood if the growth rate
of polymers can be explained by the different amount of energy transferred from
the grid and/or by the increased dissipation within the turbulent patch.
Defining a non-dimensional area growth rate,
(Apatch−At=0
patch
)· f
s (shown in Fig.
4.8) with s being the initial growth rate, s = [dA/dt]t=0, and f is the agitation
frequency, one can observe that the patches in all the cases grow similarly for the
first 5 seconds, while after 5 seconds the curves diverge, due to different dynamics
of the patch evolution.
27

0 5 10 15 20 25 30
−20
−10
0
10
20
30
40
50
60
70
80
t [sec]
(Apatch−At=0
patch)·f
s
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.8: Non-dimensional patch area Apatch. Where: f [hz] - agi-
tation frequency, s[mm2/sec] - patch growth slope shown in ﬁgure 4.4.
28

4.5 Patch equivalent radius
For simplicity of the following analysis, we define a sort of TNTI average position
in time using a patch equivalent radius, req(t):
req(t)i =
Asector (t) ·360
π · θ
(4.2)
where: θ = ±58o and Asector (t) = Apatch + Agrid where Apatch is the control
volume area defined in figure 4.3b and Agrid is the area of the agitation device
bounded by θ. This 1D approximation allows to estimate an overall patch growth
rate, which will be used later to define the entrainment rate .
Figure 4.9 depicts the equivalent radius of the patch req(t)i per each run i,
along with its ensemble average req(t) (thick curve), for the different agitation
frequencies and polymer concentrations. Although the patch radius is calculated
through the duration of the entire run, only the section up to around 10 seconds
(in the fresh water case) is used. After 10 seconds we suspect that the three-
dimensional flows affect the analysis.
For the sake of completeness, similarly to the results of the area growth rate,
we present In figure 4.10 the values of growth rate of the equivalent radius along
with the plot in logarithmic scale. The results and the conclusions are identical
to those presented in figure 4.7 . We can use these values to compare the nu-
merical exponents to a similar experimental data in the work of Wu [15]. The
main differences are in the geometry of the agitation device. Wu [15] created
a cylindrical, axisymmetric and almost 2D turbulent patch, comparing the flow
in water and a range of aqueous polymer solutions with different concentrations.
The reported value for the water case was r ∝ t0.336 (r- as defined in the article,
rms value of the patch radius) and for the polymer aqueous solutions a range of
values for the numerical exponent were reported between 0.240 and 0.147 for con-
centrations 25÷1200 wppm of poly(ethylene oxide) WSR-301 (molecular weight
Mw 4 · 106 ) . In the fully 3D flow presented in this study we measure the
req ∝ t0.18 for water case and req ∝ t0.14 for polymer solutions. Similarly to the
findings of Wu [15] the growth exponents in our experiments decrease in the pres-
ence of polymers and apparently does not depend on the concentration within this
29

0 10 20
0
20
40
60
(a)
0 10 20 30
0
20
40
60
(b)
0 10 20 30
0
20
40
60
(c)
0 10 20 30
0
20
40
60
(d)
req[mm]
0 10 20 30
0
20
40
60
(e)
0 10 20 30
0
20
40
60
(f)
0 10 20 30
0
20
40
60
(g)
t [sec]
0 10 20 30
0
20
40
60
(h)
t [sec]
0 10 20 30
0
20
40
60
(i)
t [sec]
Figure 4.9: Equivalent patch radius as measured from the center of the
agitation device and defined in equation 4.2. Color curves are five exper-
iment repetitions and the bold black curves are ensemble average. Thin
black lines show the growth slopes of the radius used in entrainment
coefficient ξ estimation using equation 4.4.
30

8 10
1
2
3
4
5
6
7
f[Hz]
(
dreq
dt)t=0[mm/sec]
0 5 10
1
2
3
4
5
6
7
CPEO[wppm]
water 6.9[Hz]
water 8.4[Hz]
water 10.5[Hz]
5ppm 6.9[Hz]
5ppm 8.4[Hz]
5ppm 10.5[Hz]
10ppm 6.9[Hz]
10ppm 8.4[Hz]
10ppm 10.5[Hz]
(a)
10
0
10
1
10
1.4
10
1.5
10
1.6
10
1.7
10
1.8
t [sec]
req[mm/sec]
water 6.9[Hz]
water 8.4[Hz]
water 10.5[Hz]
5ppm 6.9[Hz]
5ppm 8.4[Hz]
5ppm 10.5[Hz]
10ppm 6.9[Hz]
10ppm 8.4[Hz]
10ppm 10.5[Hz]
t0.14
t0.18
(b)
Figure 4.10: (a) Patch equivalent radius growth rate at t = 0 sec,
(left) versus agitation frequency, (b) versus polymer concentration.
(right)Ensemble average of equivalent patch radius (req) (see Eq. 4.2) in
the log-log scale.
32

4.6 Radial profiles of turbulent quantities
Here we present samples of representative radial profiles of enstrophy (ω2) and
kinetic energy (q = U2/2 = (u2 + v2)/2) as a function of the radial distance from
the center of the agitation device normalized by the patch radius (req it’s calcu-
lation is presented in sec. 4.5). The profiles are averaged through polar angle θ
and are defined by: ω2(r) i, q(r) i, where subscript i - stands for experiment
repetition. The spatially averaged profiles ω2(r) i are ensemble averaged for five
runs, yielding ω2(r) and q(r) , that are displayed in figures 4.11 and 4.12.
From figure 4.11 we can observe the patch growth in time (from left to right)
as it propagates in the radial direction from t = 1 sec through t = 10 sec, for all
solutions. It is also evident that the patch size in the water case is larger than
in polymer at any given moment, when comparing 5 ppm and 10 ppm we see
that lower concentration is correlated with larger patch size and stronger kinetic
energy. We also note that the peak in the water case inside the patch broadens with
time as the energy is spread through the control volume. In polymer solutions the
opposite is true where the energy grows to higher values, but does not spread in
the radial direction. This agrees with our expectations to see reduced turbulent
diffusion in polymer solution flows, in agreement with the reduced entrainment
rate, estimated below.
Comparing the kinetic energy profiles in figure 4.11 with the enstrophy pro-
files in figure 4.12 we see larger gradients of enstrophy at the interface, as expected
for TNTI (see for example ref. [14]). This fact adds to the robustness of the TNTI
identification using enstrophy. Similarly to the observations for the kinetic en-
ergy profiles, we note that for enstrophy there is a stronger peak of enstrophy for
the water case which also broadens in time. There also some non-zero enstrophy
outside the turbulent patch, although much lower than the threshold, due to weak
secondary flows inside as we saw in figure 4.1. The polymer solution profiles
show weaker enstrophy peak inside the patch and smaller patch size at all times.
The polymer solution case can be seen as very degenerate turbulent flow with
reduced turbulent kinetic energy, enstrophy and weaker entrainment.
33

10
2
10
0
10
1
10
2
r [mm]
q[mm/sec]2
water, t=1s
t=3s
t=10s
5 wppm, t=1s
t=3s
t=10s
10 wppm, t=1s
t=3s
t=10s
Figure 4.11: Spatially averaged proﬁles of kinetic energy q(r) θ at three
moments in time (1,3, and 10 sec) versus radial coordinate r for 10.5
Hz agitation frequency. All proﬁles start from the edge of the agitation
deice , r = 24 mm.
34

10
2
10
−3
10
−2
10
−1
10
0
r [mm]
ω2
s−2
water, t=1s
t=3s
t=10s
5 wppm, t=1s
t=3s
t=10s
10 wppm, t=1s
t=3s
t=10s
Figure 4.12: Spatially averaged enstrophy proﬁles, ω(r)2
θ at three mo-
ments in time (1,3,10 seconds) for 10.5 Hz agitation frequency.
35

4.7 Turbulent kinetic energy of the patch
In order to compare the entrainment properties of turbulent flows with and without
polymers , we analyze the evolution of the average turbulent kinetic energy of
the patch. As we saw from the vorticity maps (Fig. 4.1) the patch evolution in
water and the polymer solutions look very different. The control volume technique
allows to compare the flows in a relatively consistent and objective manner. Our
choice of the control volume allows to focus on the region of the flow where
turbulent diffusion is dominant and disregard the strong vertical flows along the
path of oscillation of the grid. The energy that is contained inside the control
volume is defined as:
Ei(t)CV = ρ
ˆ
CV
qi(x,y) dA (4.3)
where CV denotes the control volume defined in figure 4.3, and i subscript
stands for the index of experimental run. Equation 4.3 is implemented using a
summation rule:
Ei(t)CV = ρ
x(t) y(t)
(qi(x,y) · A)
and ensemble average of five repetitions results in an ensemble profile, E(t)CV .
In figure 4.13 Ei(t)CV and E(t)CV are plotted for different polymer concentrations
and agitation frequencies.
We recall that the agitation was turned on at t = 0 and off at t = 20 sec, hence
in Fig. 4.13 we observe a period of initial energy growth then a relatively steady
state period and a free decay of the flow after 20 seconds. Note the 10 ppm flow
where the results of all of the runs fall almost on a single curve. The steady state
period is the result of a balance between the energy injected by the grid to the
turbulence viscous dissipation within the patch and, decay due to quiescent fluid
entrainment into the patch, and fluxes through the control volume boundaries.
We should also remember that the PIV realizations provide a cross-section of the
strongly three-dimensional flow and there are some 3D effects that can affect the
results. As the flow is stronger in the water case, it is also likely that the 3D flow
36

0 10 20
0
2
4
6
8
x 10
−5 (a)
0 10 20 30
0
2
4
6
8
x 10
−5 (b)
0 10 20 30
0
2
4
6
8
x 10
−5 (c)
0 10 20 30
0
2
4
6
8
x 10
−5 (d)
ECV[Joule/m]
0 10 20 30
0
2
4
6
8
x 10
−5 (e)
0 10 20 30
0
2
4
6
8
x 10
−5 (f)
0 10 20 30
0
2
4
6
8
x 10
−5 (g)
t [sec]
0 10 20 30
0
2
4
6
8
x 10
−5 (h)
t [sec]
0 10 20 30
0
2
4
6
8
x 10
−5 (i)
t [sec]
Figure 4.13: Kinetic energy per unit length inside the control volume
calculated from PIV velocity ﬁelds. The color lines are ﬁve experiment
repetitions and the bold black line is the ensemble average. (equation
4.3)
37

effects are stronger for this case. Thus, 4.4 and 4.6 show that the patch area has
decreased after about 10 seconds in the water case whereas in the polymer the area
remains constant throughout the steady state plateau. Despite the side effects that
hinder our ability to accurately quantify the polymer dissipation contribution, we
can safely conclude that the amount of energy introduced by the grid is approxi-
mately the same (and might be even larger in the polymer case due to higher drag
coefficient), but the stronger dissipation and lower diffusion of vorticity creates a
more concentrated and less energetic patch as compared to water.
In figure 4.14 the ensemble average E(t)CV of the energy are plotted in a man-
ner where all of the different profiles are brought to the same initial level by sub-
tracting E(0)CV from E(t)CV . The evident trend of increasing steady state energy
levels as well as of initial growth slopes with increasing frequency of agitation
is observed as it is expected. What is more interesting is that the energy levels
and the initial slopes diminish as the polymer concentration is increased and the
agitation frequency is left constant. The decay phase of the profile requires a more
detailed analysis.
If we take the values of the first deflection of the profile from the steady growth
at the initial stage (ESS
CV marked by filled color circle markers on figure 4.14) and
normalize the profiles each by it’s respective value, we receive the figure 4.15. By
performing this normalization we take the agitation frequency parameter out of
the effect on energy levels. We confirm the linear proportionality ESS
CV = C · f 2
by plotting the relation in figure 4.16. We note the proportionality constant C is
different for water 5ppm and 10ppm solutions and it decreases with polymer con-
centration, so that our agitation device’s efficiency of introducing kinetic energy
into the flow in horizontal direction (analogue of the grid action parameter K in
planar space filling oscillating grid setups theory see [27]) is decreased with poly-
mer concentration. Returning to figure 4.16, also the agitation frequency should
not have an effect on the curves there is clearly a lowering energy growth rate in
the initial phase with the increase of concentration.
38

0 5 10 15 20 25 30
−2
−1
0
1
2
3
4
5
6
7
8
x 10
−5
t [sec]
ECV−Et=0
CV[Joule/m]
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.14: Ensemble average of kinetic energy per unit length inside
the control volume calculated from PIV velocity fields (equation 4.3).
The color filled circles indicate the points on the curves used to normal-
ize the data in figure 4.15.
0 5 10 15 20 25 30
−0.5
0
0.5
1
1.5
2
t [sec]
ECV−Et=0
CV
Ess
CV
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.15: Ensemble average of kinetic energy per unit length inside
the control volume (CV) calculated from PIV velocity fields and normal-
ized by the steady state energy values from figure 4.14.
39

40 60 80 100 120
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
x 10
−5
f2
[1/sec2
]
Ess
CV[Joule/m]
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
(6.01 · 10−7
) · f2
(5.91 · 10−7
) · f2
(3.96 · 10−7
) · f2
Figure 4.16: Values of total kinetic energy per unit length inside the con-
trol volume (ECV − Et=0
CV ) taken from ﬁgure 4.14 and used in the normal-
ization of ﬁgure 4.15, versus squared agitation frequency ( f 2). The thin
black lines illustrate the proportionality of the values with increasing f 2.
40

4.8 Entrainment rate coefficient
Entrainment rate coefficient, ξ represents the ratio of the growth rate of the equiv-
alent patch radius (or average TNTI radial position) to the kinetic energy that
is contained inside the control volume CV. The symbol for the coefficient is
different from the one that was used by [14] to emphasize the slightly different
definition (though they both relate to somewhat similar physical meaning). The
coefficient is defined here for the time interval of 2 to 5 seconds (through the con-
stant growth regime, but excluding the first 2 seconds of transients) (equation 4.4)
and is plotted in figure 4.17:
ξ(t) =
dreq/dt
2· ECV /(ρ· A1)
(4.4)
The variation of values of ξ within this time interval is shown in figure 4.17
and we will average the values for the comparison.
1.5 2 2.5 3 3.5 4 4.5 5
0.4
0.5
0.6
0.7
0.8
0.9
1
t [sec]
ξ(t)
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.17: Entrainment coefficient ξ values between 2 and 5 seconds
through the run.
Time averaged values of the entrainment rates we denote as ξ , and summa-
rize those in figure 4.18(left) as a function of the polymer concentration, and in
figure 4.18(right) as a function of the agitation frequency. Note that in figure 4.18
41

0 2 4 6 8 10
0.4
0.5
0.6
0.7
0.8
0.9
1
Cpolymer[wppm]
<ξ>
(a)
6 7 8 9 10 11
0.4
0.5
0.6
0.7
0.8
0.9
1
f [Hz]
<ξ>
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
(b)
Figure 4.18: (a) Estimated entrainment coefficients versus the polymer
concentration for different agitation frequencies. (b) Estimated entrain-
ment coefficients versus agitation frequency for different polymer con-
centrations. Calculated using equation 4.4 and averaged through the
values in figure 4.17.
the entrainment coefficient of polymer solutions is consistently lower than in wa-
ter case that is polymer causes the TNTI to propagate slower relative to the energy
that exists inside the patch.
In the right panel of figure 4.18 the same values of ξ are plotted versus the
agitation frequency for different concentrations of polymer. Interesting to note
how the entrainment rate coefficient grows with increasing frequency in the water
flow case, but not so in dilute polymer solution cases, where one can observe
an almost constant entrainment rate, independent of agitation frequency. It is
possibly due to the stronger reaction effect of the polymer solution to the stronger
agitation frequency within the larger patch area. The agitation increases the size
of the turbulent region in polymers, but does not change the entrainment rate at
the interface.
The entrainment rate coefficient reduction for polymer over the water case
δ = ξ polymer/ ξ water will be plotted here (figure 4.19) versus the Reynolds
number (ReM = 2π f A0 M/ν) defined by the agitation device average mesh size
M = 7 mm and the grid velocity amplitude 2π f A0 , and versus the longest relax-
ation time for the polymer chains in their respective solutions (the relaxation time
τ is a characteristic time for the polymer molecule to arrive back to an equilib-
42

800 900 1000 1100 1200 1300 1400 1500
0.3
0.4
0.5
0.6
0.7
2 π f A0 M
ν
δ
5.5 6 6.5 7
0.3
0.4
0.5
0.6
0.7
τ [msec]
δ
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.19: The entrainment rate coefficient reduction for polymer
over the water case versus grid mesh Reynolds number ReM and longest
polymer relaxation time τ calculated using eq. 3.1 in chapter 3 .
rium state after being stretched: eq. 3.1). A slight reduction in δ with increasing
Reynolds number could be recognized in the upper plot. No specific trend could
be recognized in respect to the relaxation time, which is also not too different
between the two concentrations tested in this set of experiments.
Viscosity measurements of different E-500C polymer solutions at room tem-
peratures 20 ÷ 24◦C will be summarized in the following table 4.1 as used in
Reynolds number definition above. All measurements carried out using Cannon-
Fenske size 50 capillary viscometer with efflux time of 4÷5 minutes in minimum
3 repetitions. We can safely conclude that the present experimental study reveals
the strong effect of polymers in very dilute case and solely by viscoelastic prop-
erties of the polymer, intrinsically modifying the energy transfer across the scales
of turbulent flows.
43

CPEO [wppm] ν [m2/sec]
2.5 0.83·10−6
5 0.81·10−6
10 0.85·10−6
20 1.17·10−6
50 1.22·10−6
Table 4.1: Viscosity of dilute polymer solutions.
4.9 Estimation of TNTI convoluted length
There is another important aspect that can be studied in this spatially resolved
work, the measure of the convolution or curvature of the TNTI. Quantitatively the
insight into the entrainment rate characteristics could be gained by examining the
ratio:
LT NTI
Larc i
=
T NTI
l
r(t)eq · θ
Where LT NTI is the approximate length of TNTI within the control volume
limits obtained using a sum of l , elementary length units at all the identified
grid points on the TNTI. On the other hand, Larc is the length of the arc con-
structed from the patch equivalent radius req and the control volume sector angle
θ .This ratio of two lengths is an indicator of the TNTI convolution, complexity
or curvature. It is known that where larger curvature of the interface is related
to stronger entrainment process through both engulfment and the nibbling pro-
cesses (depending on the length scale of the convoluted portions) and increased
entrainment efficiency.
In figure 4.20 we can see ensemble averaged ratios LT NT I
Larc
as a function of time
in log-log axes. We clearly see that the water case interface is of a much higher
convolution than of the polymer one, with a negligible effect of frequency . It is
noteworthy that the polymer solution case which is “closest” to the water case, for
the lowest frequency of 6.9 Hz and lower concentration of 5ppm (marked by green
diamonds) shows values higher than other polymer solutions and is comparable
44

10
0
10
1
10
0.2
10
0.3
10
0.4
10
0.5
10
0.6
10
0.7
10
0.8
t [sec]
LTNTI/Larc
water 6.9 [Hz]
water 8.4 [Hz]
water 10.5 [Hz]
5ppm 6.9 [Hz]
5ppm 8.4 [Hz]
5ppm 10.5 [Hz]
10ppm 6.9 [Hz]
10ppm 8.4 [Hz]
10ppm 10.5 [Hz]
Figure 4.20: Ratio of TNTI length to the arc acquired from req.
to water case (although smaller at all times). This observation agrees with the
behavior we have seen in figure 4.18 where the entrainment coefficient for this
polymer solution case is the highest among all other runs and the entrainment
coefficient then decreases with frequency for 5 ppm solution and lower for 10
ppm.
Also interesting to note how the ratio grows in time (especially in the water
case) as the patch develops and engulfs more and more fluid into it, in agreement
with what we observe in vorticity maps in figure 4.1 . We will discuss this and
other results in the following section, combining the inputs into a more general
insight from the presented experimental evidence.
45

4.10 Discussion
The central objectives of this work was to find out at which stage of the energy
transfer polymer could have an effect. Polymer could have an effect at the sur-
face of the agitation device by covering the surface by a thin polymer layer and
changing the roughness of the wall. Of course at any stage of energy transfer that
is to follow, specifically the turbulent energy cascade and the entrainment of the
non-turbulent fluid into the turbulent patch, polymer will have an effect. There are
no other possible sources of polymer effect on the energy content of the flow such
as submerged mechanical friction components or tank wall boundary, as could be
present in experimental setups of references: [10, 28, 14].
In appendix B we show that the power input from the steady state drag force
(which is polymer presence dependent) is one order of magnitude lower then the
power input from the added mass force (should not be influenced by the polymer
presence as soon as the concentration is low enough so as not to change the so-
lution density). Further the power input from steady state drag component was
found to be larger than in the water case. The conclusion here is that the total
energy input into the flow is at least not smaller than in the water case. Keeping
that in mind as we saw in the results chapter (figure 4.14 of CV energy content)
the amount of kinetic energy inside the patch is consistently smaller in the poly-
mer solution than in the water case and this effect is intensified by higher polymer
concentration. Hence we can objectively conclude that polymer inhibits the tur-
bulent diffusion and entrainment (figure 4.18 of entrainment rate coefficients) in
the direction of the TNTI, while the remains of the energy are injected into the
vertical flow in case of 5 wppm solution (figure 4.1 of vorticity maps) or is accu-
mulated inside two large vortexes with a higher energy density (10 wppm) than in
the water patch (figure 4.11 of energy average profiles).
Let’s consider for a moment the energy transfer mechanism from the agitation
device and into the flow in the sector of interest we analyzed in results chapter,
in water the grid creates several jets while at the same time vorticity is created at
the trailing edges of the grid deflector-profiles and released along with the jets.
The jets interact with each other in a nonlinear way producing even more vor-
ticity that constitutes the turbulent patch as we observe it in figure 4.1. What
46

we see in polymer flow is a different picture, the flow isn’t as turbulent as the
water case, it is seen also from the higher repeatability of several quantity time
profiles we saw in the results chapter (figure 4.9 for example) especially for the
higher concentration-10ppm case. The diffusion of vorticity from the two large
vortexes into the surrounding fluid is suppressed by two effects:1) TNTI shape
is smoothed by the polymer reducing the so called nibbling of the interface , 2)
A shear layer surrounding the vortexes is created which by shear sheltering(ref.
[29]) inhibits the diffusion of vorticity even more. The lower turbulent diffusion
effect in polymer could be tracked through all the figures in the results chapter,
from the smaller patch area/radius growth rate at the initial phase (figures 4.6,
4.10 and 4.5) through the smaller maximum patch area in figure 4.6 to the smaller
power law exponent in the growth phase of area and patch radius plots (figures
4.7 and 4.10), which have been observed also by Wu (ref. [15]) and to the lower
entrainment rate coefficients ξ (figure 4.18).
The convoluted length of TNTI which relates directly to the fluid entrainment
and energy transfer ability through the interface (as it is discussed in reference
[14] and [9]) was shown to be smaller in all polymer flows.
47

5 Summary and conclusions
This experimental study was devoted to the problem of turbulent entrainment
across the turbulent/non-turbulent interface with and without polymers. Study-
ing the existing literature in Chapter 2 we realize that there are many unanswered
questions regarding the effects of dilute polymers on turbulent diffusion and mix-
ing. These are seen in the strongly modified shape of the jet interface in submerged
polymer-polymer flows. In this work we tried to isolate the polymer effect to a
specific region in a flow where a relatively reliable measurements could be done.
As well we improved our measurement technique by time resolving the fluctu-
ating velocity field created by the oscillating grid, and by achieving a precise
synchronization of all experimental runs allowing an ensemble average approach.
An important development in this study was the application of a spherical grid
to improve the symmetry of the turbulent patch and allow an easy and consistent
analysis as well as a future comparison to the numerical DNS simulations by our
partners. An attempt to measure the energy input from the motor supply to the
flow was performed though proved unreliable and susceptible to large uncertain-
ties, thus a more sophisticated measurement method is required. An alternative
technique was applied (chapter B) to estimate the energy input to the flow by
measuring the steady state drag coefficient of the agitation device and using a the-
oretical model to supplement the measurements. The long process of polymer
solution preparation was automated to get more consistent results.
The experimental setup used in this study is shown in Chapter 3 and followed
by the PIV results in Chapter 4. The results are presented as vorticity maps, show-
ing the spatial and temporal evolution of the turbulent patches produced by the
spherical oscillating grid at various frequencies and solutions. The major visual
effects are the effectively lower Reynolds number flow with length scales sepa-
49

ration depletion and as a result significantly inhibited turbulent diffusion in the
direction of TNTI.
We quantified the effects using the ratio between the patch inner kinetic energy
and the radial propagation rate of the interface. Using this criterion we have found
lower entrainment coefficients in polymer flows. These results can lead to better
understanding of the polymer effects in jets, wakes and edges of boundary layers,
where polymers affect the turbulent/non-turbulent interfaces. Moreover, we could
think of the applications such as controlled drug delivery using bio-compatible
polymers (such as polyethylene glycols) that will allow us to design with better
accuracy the distance and the rate of diffusion of a given bolus with defined flow
conditions.
5.1 Future work
Future work should include the Lagrangian 3D-PTV measurements in a similar
setup quantifying equivalent flow properties while at the same time having a more
complete three dimensional picture of the entrainment interface. There is a lot
of missing analysis that can be applied to this set of data, such as structure func-
tions, spectrum and correlation functions that can help to reveal the underlying
mechanisms of the energy transfer across scales in polymer solutions.
The presented results and further analysis will be compared with the DNS
results performed by our partners from Karlsruhe Institute of Technology.
50

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54

A Experimental set A
During the study, the experimental setup was improved several times. The main
differences were the change of the grid type, from planar (see Fig. A.1) to spher-
ical (see Fig. 3.1), and the change of the polymer type, from molecular weight
of 4 (Sigma Aldrich, PEO WSR301) to 8 millions (Alkoro, E-500C) at various
concentrations. In addition, we have tried several times to estimate the energy in-
puts, from the motor to the flow (using the electronics and Labview, as described
below) and from the grid to the flow (using the drag coefficient).
A.1 Methods and materials of set A
The experimental setup that was used in set A is illustrated in following figures:
A.1. The experiments performed in a transparent glass tank filled with water or
homogeneous dilute polymer solution. A great deal of care is required for the
proper and repetitive preparation of the same polymer solution for the various
runs, later used in ensemble averaged statistics. The polymer was prepared by
manually adding small portions of approximately 1 milligram of dry polymer raw
powder at time intervals such that the previous portion is no longer visible on
the water surface. The solutions were mixed within a glass 500 ml beaker full
of filtered water. The solution was stirred at about 60 rpm by an electric stirrer
using steel propeller shaped rotor with a diameter of about half of the beaker
diameter. The solution was constantly kept at 40oC using an underneath heating
plate. The preparation process took about 48 hours to complete, depending on the
polymer concentration needed. The completion was identified by visual absence
of polymer lumps and overall visual homogeneity of the solution.
In set A the PIV experiments were performed with our standard PIV system,
55

comprising of the NEW WAVE RESEARCH dual head Nd:YAG laser (SOLO120XT)
operating at 532nm and creating a 120 mJ/pulse laser light sheet through a set of
laser lenses. Insight 3G software was controlling the synchronization of the TSI
POWER VIEW PLUS 11 megapixel camera equipped with 60 mm Nikkor macro
lens and the lasers through a dedicated synchronizer. The PIV system was not
synchronized with the vertically oscillating planar grid which was actuated by a
DC motor. The motor was controlled using a custom-made LABVIEW application
and a developed electronic circuit for the voltage/current measurements and con-
trol. Motor power supply unit had an analogue input control signal (−5 ÷ 5[V])
port, which is used to set the voltage output to the motor. The voltage supplied
to motor terminals was controlled in open loop, thus no feedback signal from the
motor shaft was used in these experiments. The system was operated by two peo-
ple, while one of them started the motor trough a LABVIEW command send to
the power control, and at approximately the same time instant one would start the
capture sequence of PIV system.
To create turbulent flow, we oscillated a small (relative to the size of the tank),
transparent grid made of PMMA was used (see figure A.1) in a center of the tank.
The grid was moved in a reciprocating motion through the water by slider-crank
mechanism, which was set to motion by a DC motor. The grid was connected
through two stainless steel rods (∼ 5.3[mm] in diameter) to slider-crank mecha-
nism, so that all mechanical parts were higher than the water level (water level
was 22 [cm] from the bottom of the tank). The grid span dimensions are much
smaller than distances between the walls of the tank, so that wall effects would be
negligible.
In two different runs we compared the flow in water vs the 20 wppm, and water
vs 50 wppm solution, each was repeated three times per three different frequencies
of the motor.
The flow was seeded using hollow glass spheres of 10 µm average diame-
ter (POTTERS INDUSTRIES INC.). The laser illuminated region was captured by
CCD camera, positioned perpendicularly to the laser sheet. Each run contained
a sequences of 50 image pairs at the frame rate of approximately 1 Hz, while
time delay between laser pulses was set to 15000 µsec (single frame exposure of
2.3 × 10−3sec and laser pulse delay of 2 × 10−3sec) . The field of view of the
56

118
709
304
509
150
230
Side View
Camera Field of view
Laser
98
389
209
Dimensions are in millimeters
Laser
Camera
Top view
Water tank
(a)
(b)
Figure A.1: (left) Preliminary experimental set-up drawing. (right)
drawing of the planar grid used in set A (dimensions in millimeters).
57

Figure A.2: Simulink model built for image analysis of vorticity fields.
(a) (b) (c)
Figure A.3: An example of image processing steps for the vorticity patch
area estimate: vorticity magnitude, blob analysis result, mask of the
blob over the gray scale map.
camera was 150 × 230 mm that was converted using the scaling of 17.5 pixel-
s/mm. Standard cross-correlation PIV method was applied with the interrogation
windows of 32×32 pixels and 50% overlap.
A.2 Patch area estimation method
Vorticity fields were first calculated from PIV vector fields using PIVMAT tool-
box and vorticity intensity maps are processed like images using image processing
tools using MATLAB/SIMULINK, see Fig. A.2.
The main steps are image loading, edge detection using Sobel filter using a
constant threshold of ω = 0.003[1/sec] , two median filters of different kernel size
(vertical and horizontal) that remove artifacts of the steel rods and the shadow of
the grid, and followed by the blob analysis that provides the area of the binary
object. The steps are shown in Fig. A.3
A.3 Preliminary results
Example flow fields can be seen on figures A.4 and A.5. It is important to mention
that in this preliminary run we had difficulties to synchronize the PIV capture with
58

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