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# Circulation in turbulent flow through a contraction

Vivek Mugundhana,b and Sigurdur T. Thoroddsena a CONTACT S. T. Thoroddsen.
Email<EMAIL_ADDRESS>Division of Physical Sciences and
Engineering, King Abdullah University of Science and Technology (KAUST),
Thuwal 23955-6900, Saudi Arabia; bDepartment of Mechanical Engineering, Amrita
School of Engineering Coimbatore, Amrita Vishwa Vidyapeetham, India.

###### Abstract

We study experimentally the statistical properties and evolution of
circulation in a turbulent flow passing through a smooth 2-D contraction. The
turbulence is generated with an active grids to reach $Re_{\lambda}\simeq 220$
at the inlet to the 2.5:1 contraction. We employ time-resolved 3-D Lagrangian
Particle Tracking technique with the Shake-The-Box algorithm to obtain
volumetric velocity fields which we use to calculate the simultaneous
circulation in three perpendicular planes. Forming a circulation vector and
studying the PDFs of the relative strength of its components, we can quantify
how the mean strain enhances and orients coherent vortical structures with the
streamwise direction. This is further studied with streamwise space and time
correlations of the circulations over a range of loop sizes. The streamwise
component of the circulation, over same-size square loops, shows increased
integral length, while the other two components are less affected. The
circulation around the compressive direction weakens and reaches prominent
negative correlation values, suggesting buckling or sharp reorientation of
transverse vortices. The PDFs of circulation transit from non-Gaussian to
Gaussian behavior as the loop size is increased from dissipative to large
scales.

###### keywords:

Contraction; grid turbulence; active grid; circulation; strained turbulence;
coherent structures; Lagrangian Particle Tracking; Shake-The-Box

##  1 Introduction

The quest for universality in the statistical description of turbulence has
lead to studies on the characteristics of velocity structure functions based
on the cascade ideas of Richardson and Kolmogorov [1]. The statistics of
circulation have been added to the arsenal used for these studies [2, 3, 4].
By virtue of the Stokes theorem, circulation statistics help in obtaining the
characteristics of vorticity, without directly computing or measuring the
vorticity field. Measurements of vorticity involve gradients of velocities,
its deductions is therefore less accurate compared to the velocity
measurements themselves. Hence there are advantages in using circulation to
characterize coherent vortical structures from experiments, which often do not
resolve the finest scales. With the advent of volumetric velocity measurements
new avenues have opened up for the study of circulation which were
inaccessible with the traditional hot-wire single-point studies of turbulence.
Herein we exploit volumentric Lagrangian Particle Tracking methods to go
beyond planar PIV to measure time-resolved circulations simultaneously in
three perpendicular planes. We do this for turbulent flow subjected to mean
strain as it is passed through a 2-D contraction, while previous studies of
circulation statistics have been focused on homogeneous and isotropic
configurations. We can thereby study how the applied mean strain enhances and
orients the direction components of the circulation vector, which we interpret
as being characteristic of coherent vortical structures. Below we list
previous studies of turbulence in contractions and then describe earlier work
on circulation in turbulence.

The contraction is an important geometry of study in turbulence which has wide
range of engineering applications, e.g. in ducting and pipe connections. Early
experiments in axisymmetric contractions are those of Uberoi [5], who
investigated effect of three different axisymmetric contraction-ratios of $c=$
${\it{4,9,16}}$, for a square cross-section duct. Following this were the
experiments by Hussain and Ramjee [6], wherein they studied the effect of
contraction wall shapes with the same $c=$ 11\. Tan-Atichat et al. [7]
performed experiments with effect of axisymmetric contractions on grid
turbulence for $c$ ranging from 2$-$36, and Ayyalasomayajula and Warhaft [8]
investigated grid-generated turbulence subjected to mean strain using an
axisymmetric contraction (4:1) in a wind tunnel. Recently, Ertunç and Durst
[9] have questioned the earlier experimental work which is based on using two-
component $\times$-configuration hot-wires. They performed experiments in
contraction with $c=$ 14.75 and showed the continuous decrease of streamwise
fluctuations, by carefully accounting for possible errors involved in hot-wire
measurements. Brown et al. [10] used two-component Laser-Doppler Velocimetry
(LDV) to study turbulence in a planar contraction. Thoroddsen and Van Atta
[11] investigated the effect of a 2-D contraction ($c=$ 2.5) in a thermally-
stratified wind tunnel, with a strong linear temperature gradient. In their
subsequent work [12] they studied turbulence in a constant-area vertical
expansion.

Circulation over a closed contour $C$ cutting through a flow domain, is
obtained by integrating the velocity vector field $\bf{u}$ projected onto the
tangent of the curve along the closed contour, given by

$\Gamma_{A}=\oint_{C}{\bf{u}}\cdot dl=\oint_{A}{\boldsymbol{\omega}\cdot dA}.$
(1)

Here, $A$ is the area enclosed by the closed contour $C$ and
$\boldsymbol{\omega}=\nabla\times\bf{u}$ is the vorticity field. As the
contour can be characterized by a linear dimension $r$, we also denote
$\Gamma_{A}$ calculated by Eq. (1) as $\Gamma_{r}$. In this equation, $dl$ and
$dA$ represent the contour elemental tangent length and the elemental area
enclosed by the contour respectively.

The earliest studies of circulation in turbulent flows has been by Migdal [2]
and Sreenivsan et al. [3, 4]. Some of the main characteristics of the
probability density function (PDF) of circulation obtained in homogeneous-
isotropic turbulence are: (i) the PDF of circulation satisfies the area rule
[2] i.e, its form is only a function of the size of area enclosed by the
contour but not the contour’s shape; (ii) by the law of large numbers, the PDF
should be Gaussian for large values of $A$, given that the dimensions of the
contour are in the inertial range (IR). Another aspect of interest is the
scaling of circulation with $r$ and their comparison with Kolmogorv’s scaling
of the velocity increments. According to Kolmogorov theory [2, 13], the $n$-th
moment of circulation scales with $r$ as,

$\left\langle{\left(\Gamma_{A}\right)}^{n}\right\rangle\sim
r^{\lambda_{n}}\sim r^{4n/3}.$ (2)

As $A=r^{2}$, we can write this scaling as,
$\left\langle{\left(\Gamma_{A}\right)}^{n}\right\rangle\sim
A^{\lambda_{n}/2}$. This scaling is strictly applicable when the circulation
PDFs are Gaussian distributed.

The circulation properties and their scaling have been investigated in
experimental [3, 14, 15], numerical simulations [16, 17, 18, 19, 20] and
theoretical studies [2, 21]. Moriconio and Takakura [21] obtained statistics
using saddle-point based Martin-Siggia-Rose (MSR) technique. Migdal [2] first
started studies on the statistical properties of circulation. He proposed that
the circulation PDF depends on the area of the loop but not the shape. Umeki
[16] showed that the circulation second moment for a figure-eight loop (loop
consisting of two square loops with one common vertex) scales with the scalar
area of the constituent loops, for 2-D decaying isotropic turbulence. He
suggested that the area rule be called scalar-area rule and modifications to
the functional forms used in the circulation statistics to match the observed
behavior. Benzi et al. [17] also showed that the circulation PDF depends on
the area and not the shape for 3-D shear flows using his simulations at
Taylor-Reynolds number Re${}_{\lambda}\sim$ 40\. Recently, Iyer et al. [19]
showed that the PDF of circulation is independent of dimensions of rectangular
loops over the entire PDF even at very high Reλ using DNS, which goes against
Migdal’s arguments that area rule is applicable only for the tails. They
performed high resolution simulations for Reλ in the range 140 to 1300 using
meshes up to $16384^{3}$. At higher Reλ, area rule becomes applicable over
wider range of scales in the IR. They also showed that the circulation
computed over a figure-eight loop scaled with the scalar sum of area of the
loops. In their subsequent work [20], they test the validity of this area rule
for non-planar loops. For non-planar loops, the area taken is the minimal
surface enclosed by the loop. The PDFs of circulation differed for planar and
non-planar loops having the same area and hence the area rule is not true.
However, the PDF of the circulation normalized by the 8-th moment, shows a
good collapse for both planar and non-planar loops.

The circulation PDF is Gaussian for large loops with its dimension in the IR,
however deviates from the Gaussian behavior as the loop size tends to the
dissipation scales. Sreenivasan et al. [3] reports this in their seminal early
study wherein they explored the circulation properties in the turbulent wake
of a cylinder at Re${}_{\lambda}\leq$ 40, using two-dimensional particle image
velocimetry (PIV). They observed that the PDF deviated from Gaussian and
approached exponential form for smaller loop areas. Later, Zhou et al. [14]
experimentally measured the PDFs and the moments of velocity and circulation
structure functions in turbulent Rayleigh-Bénard convection in a cylindrical
convection cell using PIV technique. The measurements were carried out in
central region of the cells, in a vertical plane, far from the boundaries and
hence they approximated the flow to be homogeneous and isotropic. They showed
the flatness of the PDF tended to the Gaussian value 3 with increase in the
characteristic dimension. Thoroddsen [15] presented an approach based on
conditional averaging, to study the spatial structure of dissipation in the
context of validation of Kolmogorov’s refined similarity hypothesis. He
presented the PDF of a large-scale vorticity about the transverse direction
for grid turbulence at Re${}_{\lambda}=$230\. The vorticity is calculated
using velocities measured across finite distances using multiple probes
consisting of 4 single-wire and one X-wire probe. This quantity being computed
over finite distance can be viewed as circulation of loop with a
characteristic dimension to be of the order of the probe spacing. In line with
this analogy, he reported the large-scale vorticity is nearly-Gaussian
distributed with a flatness of 3.2.

The transition of circulation PDF from non-Gaussian to Gaussian has also been
observed in direct numerical simulations (DNS) [4, 16, 17, 18, 19, 20]. Umeki
[16] reports this behavior at Re${}_{\lambda}=$ 100 in 2-D decaying isotropic
turbulence. Cao et al. [4] showed this for a more isotropy-controlled
homogeneous-isotropic turbulence at $Re_{\lambda}=$ 101, 181 and 216. They
showed that the flatness of the PDF tends to Gaussian value of 3, as the
contour approaches the integral scale, but remains higher than that of the
velocity increments, indicating circulation to be more intermittent. This
higher intermittency of circulation PDF was also reported by Yoshida and
Hatakeyama [18] in their DNS data at $Re_{\lambda}=$ 90\. Using a threshold
value for vorticity magnitude, they decomposed vorticity into strong and weak
components. Circulation computed based on the weak vorticity showed a self-
similar behavior with loop size and was nearly Gaussian. The flatness of the
distribution was close to 3 when threshold vorticity magnitude is chosen to
twice its rms value. On the other hand, circulation associated with the strong
component showed stretched tails and departure from the self-similarity. They
concluded that the circulation associated with strong-vortical activity
contributed to the non-Gaussian behavior. They proposed a model which matches
the asymptotic behavior of the strong component circulation.

Recent studies by Iyer et al. [19, 20] showed that for rectangular loops of
different aspect ratios, but the same area, the circulation PDFs collapse into
a single curve, for loops contained inside the IR. However, this fails when
one of the dimension lies outside the IR. It is to be noted that unlike the
previous studies, they present PDF for circulation $\Gamma$ normalized by the
product of root-mean-square velocity fluctuation and the integral length
scale. They showed that the flatness of the distribution is 3 at large scales,
increases to a higher value and remains fairly constant in the IR. At small
scales, the flatness value approaches its dissipation-limit which is given by
the much larger flatness values of the vorticity component normal to the
circulation plane. They observed that for the highest Reλ, the circulation
flatness remained high, but constant with loop size in the IR, which indicates
lower intermittency of circulation. By extrapolation, they predicted the
intermittency-free limit to be at Re${}_{\lambda}=$ 1900\. One of the
important finding in this work is the scaling properties of circulation’s
moments. The moments have a simpler bifractal behavior - (a) lower moments
$n<3$, $\lambda_{n}=$ 1.367$n$ scaling which is very close to the Kolmogorov
scaling, and (b) for $n>3$ (up to 10) has sub-Kolomogorov scaling with a
fractal dimension of 2.2.

Circulation statistics have attracted attention in turbulence after the
theoretical work of Migdal [2], following which Sreenivasan and his co-workers
[3, 4, 19, 20] have explored the characteristics of circulation with
experiments and DNS. The major issue that has been in focus is the validity of
the area rule. However, we see from this literature survey that most of the
studies have focused on the idealized isotropic turbulence and the
experimental studies are based on two-dimensional PIV techniques, providing
$\Gamma$ only in one plane. The flow measured in these studies, which include
wakes and thermal-convection cells which are far from isotropic on the large
scales. In the current study, we look at circulation statistics in a non-
canonical turbulent flow - strained flow through a smooth contraction. We
employ the state-of-the-art  time-resolved 3-D Lagrangian Particle Tracking
(LPT) technique using high-speed cameras to track particle trajectories. The
particle velocities are then mapped on a grid in volumetric slice to obtain
the Eulerian velocity field. This enables us to present circulation over loops
contained in mutually perpendicular planes, and such results have not been
reported. By virtue of the measurement technique, we present the relative
strengths of the circulations computed on the three mutually-perpendicular
planes with center aligned with the center of the contraction. Such 3-D
measurement techniques make it possible to compute circulation in non-planar
and complex loops and hence verify the limits of the area rule. The effect of
straining on the evolution of circulation statistics like its rms and PDF are
presented. We obtain the space and space-time correlations of the circulations
and deduce from there an integral-scale based on the spatial correlation. The
moments of the PDFs and the scaling exponents for varying size of loops are
also presented.

Figure 1:  Measurement regions are shown with respect to the coordinate axis
positioned at the Start of Contraction ($SOC$). $EOC$ denotes the End of
Contraction. Regions P1 and P2 approximately measure 106 mm ${\times}$ 106 mm
and 56 mm ${\times}$ 106 mm respectively. Here $x_{AG}$ represents the
distance from the bottom shaft of the active grid. The flow is along the
positive $x$ coordinate.

##  2 Experimental methodology

###  2.1 Experimental setup

We study the evolution of circulation through a smooth 2-D contraction
(contraction ratio, $c=$ 2.5) in a constant-head vertical water tunnel. The 2D
schematic of the contraction with the measurement regions is shown in Figure
1. Water from the overhead tank flows through a straight vertical inlet duct
of cross section 260 mm${\times}$ 260 mm, where the background turbulence is
suppressed with flow conditioners. Following this, the flow is guided smoothly
into the active grid module through a 203-mm long converging section which
reduces the cross section to 180 mm${\times}$ 180 mm. The active grid with
spacing of $M=$ 30 mm greatly enhances the turbulence level. Downstream of the
grid, water flows through a 240-mm straight section which homogenizes the flow
before it enters the contraction. The $180$-mm long contraction smoothly
reduces the cross section from $180\times 180$ mm to $72\times 180$ mm.
Constant head is maintained in the overhead tank using a centrifugal pump. The
level in the overhead tank and thus the flow rate through the tunnel is
controlled using ball valves, after the pump. All experiments were performed
with the valve set to $\left\langle U_{in}\right\rangle\approx$ 0.3 ms-1 at
the inlet to the contraction. A detailed description, illustrations of the
experimental setup and the active grid used in this study can be found in our
previous work, Mugundhan et al. [22]. Note that, the modified setup with a
240-mm extension inserted between the grid and the contraction, as discussed
in [22], is used in this study.  All measurements are performed 5 minutes
after a steady flow is achieved in the loop. The experiments are performed at
21∘C, giving water density of $\rho=$998 kg/m3 and dynamics viscosity of
$\mu=$9.79$\times$10-4 Pa-s.

For this study, we choose two grid rotation protocols $-$ the synchronous mode
1 (S1) and the random mode (R). Of the modes employed in [22], it is seen that
modes S1 and R injected the highest level of turbulence. In mode S1, all
shafts rotate synchronously at a constant speed of $N=$4 rps (Rossby number,
$Ro=2\left<U_{in}\right>/M\Omega\approx$0.80, where $\Omega=2\pi N$) with
alternate shafts rotating in the opposite direction. In the random mode, the
shafts are rotated in the single-random protocol at a constant speed of
$N=$3.5 rps ($Ro\approx$0.91). The cruise time varies randomly between the
period of 180∘ to 540∘ of a rotation. This ensures that the average cruise
time equals the period of one complete revolution. The random mode results in
the highest Taylor Reynolds number, of $Re_{\lambda}\sim 220$ at the start of
contraction.

###  2.2 Imaging, calibration and seeding particles

We use the LaVision Tomo-PIV imaging system to acquire the particle images and
obtain volumetric flow field. Particles are illuminated using a dual-cavity
Nd-YLF, 527 nm green laser and LaVision volume optics (see [22] for details on
the illumination setup). We use the 3-D LPT technique with the shake-the-box
(STB) algorithm [23] to track the particles. The particle velocities are then
mapped on to a volumetric grid to obtain the three-dimensional Eulerian
velocity field. Region P1 is inside the contraction, with its top portion 5 mm
above the start of the contraction (SOC), and region P2 is at the bottom of
the contraction, which includes the exit and the following straight section.
The volumetric regions P1 and P2 measure $106\times 106\times 24$ mm3 and
$56\times 106\times 24$ mm3 respectively. Even though the illumination volume
was 24 mm thick, only the velocity calculations on a grid of thickness
$\approx$ 18$-$19 mm was used in the statistics. The spurious vectors seen
occasionally close to the edges were therefore not included in the analysis.

Imaging is done using four high-speed video cameras (Phantom v2640) and 85 mm
Nikkor tilt lenses at an aperture of f/22. The tilt lenses are employed to
ensure that a greater depth of the measurement volume remains in focus. The
cameras can go up to 6600 fps at full 2048$\times$1952 resolution. The
arrangement of the cameras is similar to that used in the measurements in
[22]. Cameras 1 & 2 are placed on one side of the contraction viewing its
front side, while Cameras 3 & 4 are placed on the opposite side. For good
reconstruction quality, we maintain optimum angles between the cameras.
Cameras viewing from opposite sides are positioned so that they do not have
co-linear lines of sight. For position P1, the angles between C1/C2 and C3/C4
are 32∘ and 30∘ respectively, while for the narrower section in P2 they are
29∘ and 27∘. In P1, cameras C1, C2, C3, C4 make angles 106∘, 74∘, 75∘ and 105∘
with the laser respectively. In region P1, images are captured with a
resolution of 2048$\times$1952 px at 1300 fps. However, in position P2 at the
contraction exit, images are captured at a lower resolution of
1024$\times$1952 px at the same 1300 fps. Imaging at P1 and P2 require major
changes in the laser optics and the cameras. Hence, experiments in the two
measurement regions are run on separate days.

Spatial calibration is done using two 11.8 mm thick 4-plane calibration plate
from LaVision (Number: 106-10). The whole calibration plate of size 106 mm
$\times$ 106 mm was used for spatial calibration in region P1. For the smaller
region P2, a modified plate with a reduced size of 56 mm $\times$ 106 mm was
used. Calibration is performed with the test section filled with water and
normal lighting. Third-order polynomials that map the physical coordinates to
the image coordinates are obtained using the LaVision DaVis software. The
calibration is followed by volume self-calibration, which accounts for any
misalignments such as slight changes in the position of cameras and other
optical disturbances during the experiment. We use Tween-80 surfactant-treated
fluorescent red or orange polyethylene microspheres (from Cospheric) as
seeding particles. The particles have diameters in the range $63-106$
${\mu}$m, which are close to neutrally buoyant with a density of 1.05 g/cm3
which gives a Stokes number of 2${\times}$10-4.

###  2.3 Velocity computation using STB

We use shake-the-box (STB) algorithm for the velocity computations. STB is a
4-D Lagrangian particle-tracking velocity measurement algorithm that can
handle high seeding density in the flow [23, 24]. The raw images acquired by
the cameras are pre-processed by subtracting a sliding minimum over 5 pixels
and normalizing intensities with a local average. In the next step, the volume
self-calibration is done to improve the original calibration using the pre-
processed images. This step accounts for any misalignments in the camera
positions and other optical disturbances that could occur during the conduct
of the experiment. The STB algorithm tracks the particles in time and hence
makes the Lagrangian velocities available, which is then mapped on to a
regular Eulerian grid. The three-dimensional vectors are smoothed over
3$\times$ voxels for spatial and time continuity of the vectors. We use the
the STB algorithm implemented in the LaVision’s DaVis (version 10.1.1)
software in this study. The algorithm results in about 115,000 and 30,000
tracks in regions P1 and P2 respectively. We use velocities mapped on a grid
of size 48$\times$48$\times$48 voxels with an overlap of 75% and a time filter
length of 5 time steps for the circulations statistics. We use 3-D velocity
grids of size 176$\times$165$\times$33 & 57$\times$168$\times$35 in regions P1
and P2 respectively. This gives us 0.96 M vectors & 0.34 M vectors in regions
P1 and P2 respectively. On an average there are 10$-$15 particles identified
per 48$\times$48$\times$48 voxels which is used for velocity calculations. The
particle concentration varied between 0.03$-$0.06 particles-per-pixel. The
grid resolution with this setting is $\delta$= 0.64 mm, which $\sim$ 2.1 $-$
2.4 times the smallest Kolmogorov scale. This choice of grid resolution was
based on our systematic study on the effect of resolution and time filter
length on the velocities in [22]. The uncertainty in the velocity measurements
and the convergence of our statistics are included in Appendix E.

Parameters | Sync Mode 1 | Random
---|---|---
${{\left\langle u^{2}\right\rangle}^{1/2}/\left\langle U_{in}\right\rangle}$ ${{\%}}$ | 5.3 | 6.4
${k}$ (m2 s-2) | ${4.83\times 10^{-4}}$ | ${6.12\times 10^{-4}}$
${\varepsilon}$ (m2 s-3) | ${1.76\times 10^{-4}}$ | ${1.20\times 10^{-4}}$
${L}$ (mm) | 14.49 | 25.80
${\lambda}$ (mm) | 5.98 | 8.65
${\eta}$ (mm) | 0.271 | 0.298
${Re_{L}}$ | 325 | 652
${Re_{\lambda}}$ | 134 | 218
$\delta/\eta$ | 2.4 | 2.1
$S^{*}=Sk/{\varepsilon}$ | 2.5 | 4.4

Table 1: Turbulence flow parameters at the start of contraction
($x_{AG}/M=16$). Here ${\left\langle U_{in}\right\rangle}$ is the mean inlet
velocity, $u$ is the streamwise velocity fluctuation, ${k}$ is the turbulence
kinetic energy, ${\varepsilon}$ is the dissipation rate, ${L}$ is the integral
length scale, ${\lambda}$ is the Taylor microscale based on the streamwise
correlation function $f$, ${\eta}$ is the Kolmogorov length scale, ${Re_{L}}$,
${Re_{\lambda}}$ are Reynolds numbers based on $L$ and ${\lambda}$, $\delta$
is the velocity-grid resolution and $S^{*}$ is turbulence-to-mean-strain time
ratio. Figure 2: The mutually perpendicular square loops used to compute
circulations about point $P$. The loops shown in the left, middle and the
right are used to calculate $\Gamma_{x}$, $\Gamma_{y}$ and $\Gamma_{z}$
respectively. A cubical 5$\times$5$\times$5 Eulerian grid is shown in the
figure as an example and the circles represent the velocity locations on the
grid. The arrows along the side of the loop indicate the direction of
integration. The stretching and compression directions are indicated by the
bold arrows. The dotted lines indicate a deformed fluid element which
initially matches with the square loop.

###  2.4 Turbulence parameters

The important turbulence parameters computed for a point on the centerline and
at $SOC$ ($x=$ 0) is tabulated in Table 1. The notation used in this paper is
the following: the instantaneous velocity $U=\left\langle U\right\rangle+u$,
$V=\left\langle V\right\rangle+v$, $W=\left\langle W\right\rangle+w$, where
$\left\langle U\right\rangle$, $\left\langle V\right\rangle$ and $\left\langle
W\right\rangle$ are the time averaged velocities and, $u$, $v$ and $w$ denote
the fluctuations. The values presented are averaged over independent
realizations. In each mode, 4 cases are recorded in region P1 and 6 cases in
region P2. Each realization occupied the full memory of the four cameras
($\approx$ 1 terabyte).

The mean streamwise velocity at the inlet to the contraction varied between
0.29 and 0.31 ms-1 in the experiments. As seen in our previous work, the
random mode results in the highest $Re_{\lambda}$ which is $\sim$ 220\. The
lengths scales presented are based on the spatial velocity correlation
function $f$. The Taylor microscale estimated as
$\sqrt{u^{2}/{\left\langle\partial u/\partial x\right\rangle}^{2}}$ [25] gives
slightly smaller values for $Re_{\lambda}$. Using this estimate, the largest
of our $Re_{\lambda}$ at the start of contraction, which occurs for the random
mode, reduces from 218 to 170.

| Synchronous Mode | Random Mode
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 3: Streamwise variation of circulation r.m.s along the centerline for
synchronous mode S1 (left) and the random mode R (right). Circulations
computed in the perpendicular planes: (a) $\Gamma_{x,rms}$ (Straining
direction); (b) $\Gamma_{y,rms}$ (Compression direction); and (c)
$\Gamma_{z,rms}$. $\left\langle U_{in}\right\rangle$ is the the area-averaged
mean inlet velocity in the straight section and $M$ is the mesh size.

##  3 Results and Discussion

###  3.1 Streamwise variation of circulation

We present the effect of straining on the circulation r.m.s in this section.
We use the velocity field obtained from the STB calculations to determine the
circulation by integration along a square loop of dimension $r$ and centered
about a point $P$ as shown in Figure 2. The planes are shown in a subset of
the regular Eulerian grid consisting of 5$\times$5$\times$5 grid points. The
nomenclature used for circulation is based on the unit normal of the loop. For
example, the circulation calculated by performing the line integration (Eq. 1)
of the velocity vector along the square loop on the extreme left in Figure 2
is denoted as $\Gamma_{x}$, with $-x$ being the normal direction. Note that
the flow through the contraction is along the $+x$ direction. While
calculating $\bf{u}\cdot dl$ over the line element $dl$ connecting two
adjacent nodes, average of these velocities is used. The circulations
$\Gamma_{x}$, $\Gamma_{y}$ and $\Gamma_{z}$ may be viewed as the rotation
strength of large structures of dimensions comparable to the size of the loop
. In the limiting case of the loop size shrinking to point, the circulation in
these planes represents the corresponding components of the local vorticity.

Figure 3 shows the streamwise variation of the r.m.s circulation
$\Gamma_{x,rms}$, $\Gamma_{y,rms}$ and $\Gamma_{z,rms}$ for the synchronous
and the random modes. The r.m.s values are normalized using the mean velocity
at the inlet to the contraction $\left\langle U_{in}\right\rangle$ and the
mesh size $M$. The results shown for a particular measurement region is the
average of five realizations. Circulation is computed for four square loops
centered about the centerline of the contraction and having sizes $r=$ 2.6 mm,
5.1 mm, 8.9 mm and 10.2 mm, which cover the range of scales from near
dissipative to large scales. This corresponds to a $r/\eta=$ 5 $-$ 40\. Note
that the Taylor microscale is in the range 5 $-$ 9 mm.

(a) | (b)
---|---
|
Figure 4: Streamwise variation of (a) the streamwise velocity $\left\langle
U\right\rangle$, and (b) the normalized mean strain $\partial\left\langle
U\right\rangle/\partial x$ along the centerline, for the synchronous and
random grid-rotation modes. Here $\left\langle U_{in}\right\rangle$ is the
mean velocity at the inlet to the contraction and $M$ is the grid mesh size.
The vertical line marks the end of contraction.

We see a slight discontinuity in the profiles between the measurement regions.
The measurements in regions P1 and P2 were made with different optical/camera
set-up and alignment on different days with major changes in the optical
system, camera orientation and different calibrations. As pointed out in [22],
the transverse inhomogeneity from the active grid and the mismatch of
illuminated volumes depth could be another reason for the discontinuity in
profiles. However, the variation is similar to that seen with the r.m.s
vorticity. We presented the streamwise variation of all three components of
velocity and vorticity for the same contraction in our earlier study [22]. In
Figure 3 we see that the circulation $\Gamma_{x,rms}$ increases with $x$
initially up to $x\approx$ 150 mm, following which it remains approximately
constant to $x\approx$ 175 mm. On the other hand, $\Gamma_{y,rms}$ decays till
its minimum at $x\approx$ 175 mm, but then shows an increase until the exit.
This is because of the mean strain experienced by the vortical structures. The
streamwise variation of the mean velocity and the dominant mean strain
$\partial\left\langle U\right\rangle/\partial x$, along the centerline of the
volume, is shown in Figure 4. The flow accelerates through the contraction
reaching 2.5 times the inlet velocity at $x=$ 180 mm as seen in Figure 4(a).
The mean strain increases, reaching a maximum at $x=130$ mm, following which
it decreases till the end of the measurement region as seen in 4(b). The
contour and vector plots of the time-averaged velocity fields in measurement
regions P1 and P2 are shown in Figure C1. The vortex tubes aligned along the
flow direction gets stretched by the mean strain, whereas those along the
transverse direction get compressed by the conservation of angular momentum.
This explains the increase in $\Gamma_{x,rms}$ and decrease in
$\Gamma_{y,rms}$. The dip observed in $\Gamma_{x,rms}$ and $\Gamma_{y,rms}$ is
at $x\approx$ 175 mm slightly before the exit of the contraction which is at
$x=$ 180 mm. Hence, we think that the maximum strain that occurs at $x\approx$
130 mm, does not coincide with the dip, which may occur where the return to
isotropy starts at the end of the contraction. The subsequent increase for
$x>$ 180 mm following the exit of the contraction, indicates return to
isotropy. This increase towards the exit is prominent in $\Gamma_{y,rms}$ when
compared to $\Gamma_{x,rms}$. Weak contractions are sometimes used after the
turbulence-generating grids to improve isotropy [26, 5]. Uberoi [5] in his
experiments with 4:1 axisymmetric contraction reports the tendency of
streamwise r.m.s velocity $u_{rms}$ to increase towards the transverse r.m.s
velocity $v_{rms}$ after exiting the contraction. Figure 3(c) shows the
circulation around the $z-$axis $\Gamma_{z,rms}$ is relatively constant along
the flow direction because of the complete lack of mean strain in the $z$
direction in our 2-D contraction.

The profiles obtained with different loop sizes are qualitatively the same
except that the larger loops result in higher r.m.s values. The circulation
strength increases with loop area as circulation is the area integral of
vorticity normal to the area of integration. With larger loops, circulation
accounts for a large-scale rotation and hence results in higher values of
fluctuations. It can also be seen that the initial grid protocol has minimum
effect on the circulation r.m.s variation. However, random mode shows slightly
higher r.m.s values as it is seen to inject higher levels of turbulence
compared to the synchronous mode.

|  $r\sim$ 25$\eta$ |  $r\sim$ 50$\eta$
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 5: The streamwise variation of probability density function (PDF) of
the relative circulation strengths of $\Gamma_{x}/\left|\Gamma\right|$ (a),
$\Gamma_{y}/\left|\Gamma\right|$ (b) and $\Gamma_{z}/\left|\Gamma\right|$ (c)
measured with the random mode, loop sizes $r\sim$ 25$\eta$ (left) and $r\sim$
50$\eta$ (right). The circulation strength, $\left|\Gamma\right|$ is
calculated as
$\left(\Gamma_{x}^{2}+\Gamma_{y}^{2}+\Gamma_{z}^{2}\right)^{0.5}$. The
streamwise variation is shown near the start of contraction $(SOC)$ at $x=$ 8
mm, near the maximum straining location at $x=$ 130 mm and after the end of
contraction $(EOC)$ at $x=$ 190 mm. The PDF is computed using 50 bins. At the
inlet (red curves) all the three circulation components are evenly distributed
while inside the contraction the circulation aligns in the streamwise
stretching direction and it’s strength in the compression direction becomes
small.

###  3.2 Relative strength of circulation components

We can consider the circulations in the three perpendicular planes, as the
components of a ‘vector’. The relative strength of these circulations then
gives indication of the local orientation of coherent vortices. In Figure 5 we
present this in terms of PDF of relative strengths of components and how they
change in the streamwise direction. The strengths of these circulation
components are normalized by its instantaneous absolute value given by

$\left|\Gamma\right|=\sqrt{\Gamma_{x}^{2}+\Gamma_{y}^{2}+\Gamma_{z}^{2}}$ (3)

This is computed at three streamwise locations, first near the inlet ($SOC$),
at the maximum straining location and after its end ($EOC$).

Circulation computed over a closed contour is the area-integral of the
vorticity enclosed by the contour. Hence the relative strength of the
circulation components can be viewed as the instantaneous orientation of the
vortical tube/filament enclosed by the contour. A single vortex tube aligned
exactly with the streamwise direction would give
$\Gamma_{x}/\left|\Gamma\right|=\pm$ 1 and zero for the other two components.
Sreenivasan et al. [3] have argued that circulation can take adequate account
of the contribution of filament-like structures unlike structure functions,
which are just one-dimensional cuts of the three dimensional turbulence.

The left and right columns of Figure 5 show the PDF of the relative strength
computed with the random mode, for two loop sizes, in the IR, of $r\sim$
25$\eta$ and $\sim$ 50$\eta$ respectively. We see at the inlet (red curves)
that the orientation is random with uniform PDF for all three components of
$\Gamma$. In contrast, as we move through the contraction the streamwise
circulation PDF, in Figure 5(a), tends to peak at $\pm$1 and this peak value
increases slightly downstream from $x=130$ to 190 mm. This indicates greater
relative strength of the streamwise circulation ($\Gamma_{x}$) compared to the
circulation in the other two planes. This also implies the preferential
alignment of coherent vortical structures in the $x-$direction. The transverse
compression of the contraction reduces the strength of $\Gamma_{y}$,
eventually making $\Gamma_{y}\simeq 0$ more probable at the end of the
contraction (blue curve), as seen in Figure 5(b). On the other hand, the
distribution of $\Gamma_{z}/\left|\Gamma\right|$ undergoes little change as we
move along the contraction as seen in Figure 5(c). However, for the smaller
loop $r\sim$ 25$\eta$, $\Gamma_{z}$ is only slightly less uniform compared to
that for $r\sim$ 50$\eta$. The circulation tends to be more dominant in the
$x$ direction, rather than the transverse $y-$ or $z-$directions. At the inlet
(red curves) all the three circulation components are approximately evenly
distributed while inside the contraction the strongest circulation aligns in
the streamwise stretching direction and it’s strength in the compression
direction becomes small. For similar conditions, we showed in [22] that this
preferential alignment exists with the vorticity vector and the coherent
vortical structures identified based on the vorticity magnitude.

The PDFs in Figure 5 are close to symmetric from the beginning of the
contraction, as is expected with the random mode changing the grid-rotation
direction, yielding a homogeneous distribution. On the other hand, the
synchronous mode’s PDFs exhibit a significant asymmetry. The results for the
synchronous mode are therefore relegated to the Appendix section.

To further characterize this non-homogeneity we present the 2-D contour plots
of the average and r.m.s vorticity for both the synchronous and random modes
in Appendix Figures B1 and B2. The contours are shown in the mid $x-y$ plane
of the measurement region P1. We see much ‘better’ transverse homogeneity with
the random mode when compared to the synchronous mode. However, keep in mind
that the color is selected to highlight rather small non-homogeneities in the
mean fields.

|  $x=$10 mm |  $x=$130 mm
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 6: The spatial correlation of circulations in the three perpendicular planes: (a) $R_{\Gamma_{x}}$, (b) $R_{\Gamma_{y}}$, and (c) $R_{\Gamma_{z}}$ for mode S1 and different loop sizes. The correlation is computed between points on the centerline of the measurement region by varying their streamwise separation $\Delta x$ with $x_{1}=$10 mm (left) which is near the $\it{SOC}$ and at $x_{1}=$ 130 mm (right) which is downstream of the maximum strain location. | Synchronous Mode | Random Mode
---|---|---
(a) |  |
(b) |  |
Figure 7: The variation of the circulation integral scale $L_{\Gamma}$ with
loop size. The integral scale corresponds to the correlations obtained with
the first points at $x_{1}=$10 mm (a) and at $x_{1}=$130 mm (b). The integral
scale is computed by integrating the absolute values of the spatial
correlation functions ($R_{\Gamma_{x}}$, $R_{\Gamma_{y}}$, $R_{\Gamma_{z}}$)
from $\Delta x=$ 0 to the point which corresponds to the second zero
crossover.

###  3.3 Integral Length Scale of $\Gamma$

In turbulence, correlation functions of velocity components separated in space
or in time are used to characterize the associated length and time scales. In
line with this, we define a correlation function $R_{\Gamma}$ which correlates
circulation around two central points separated by $\Delta x$ in the
streamwise $x-$direction only, for example, the correlation function for
$\Gamma_{x}$ is defined as

$R_{\Gamma_{x}}(x_{1})=\frac{\left\langle\gamma_{x}(x_{1},t)\gamma_{x}(x_{1}+\Delta
x,t)\right\rangle}{\left\langle\gamma_{x}(x_{1},t)^{2}\right\rangle},$ (4)

where the instantaneous $\Gamma_{x}(x,t)$ is decomposed into its mean and
fluctuation components,
$\left\langle\Gamma_{x}(x)\right\rangle+\gamma_{x}(x,t)$. The correlation
functions $R_{\Gamma_{x}}$, $R_{\Gamma_{y}}$ and $R_{\Gamma_{z}}$ are
calculated having the first point $x_{1}$ fixed and by varying the parameter
$\Delta x$. To understand the effect of straining, we compare these
correlations at $x_{1}=$ 10 mm, which is just inside the contraction and at
$x_{1}=$ 130 mm, which is slightly downstream of the maximum straining
location. We present the correlations over a wide range of square-loops sizes
between $\approx$ 5$\eta$ and 60$\eta$ for the synchronous mode in Figure 6.
The space shift $\Delta x$ is always in the $x$ direction and hence
$R_{\Gamma_{x}}$ is a longitudinal correlation whereas $R_{\Gamma_{y}}$,
$R_{\Gamma_{z}}$ are the transverse cross-spatial correlations. We see that as
$\Delta x$ increases, the initially positively correlated circulation crosses
over to negative values. The first crossover happens over a shorter $\Delta x$
with $\Gamma$ over small loops when compared to $\Gamma$ over large loops. For
example, the crossover of $R_{\Gamma_{x}}$ occurs at $\Delta x\approx$ 15 mm
with $r=$ 4.7$\eta$ and at $\Delta x\approx$ 28 mm with $r=$ 61.5$\eta$ (see
Figure 6(a) on the left side). This is true with all the three correlation
functions. As pointed out earlier, $\Gamma$ computed over small loops
approaches the behavior of vorticity. On the other hand, $\Gamma$ over large
loops encompasses the vorticity over the loop area and hence is representative
of large coherent structures. Hence, $R_{\Gamma}$ over larger loops exhibit
positive correlation over a longer distance. In certain cases, we do see an
early second crossover of the function to positive values. This second
crossover is very evident with $R_{\Gamma_{y}}$ and it occurs over a small
$\Delta x$ as seen in Figure 6 (b). It is interesting to note that the first
crossover point of $R_{\Gamma_{y}}$ at $x=$ 130 mm approximately matches the
loop size $r$. The crossovers are observed at $\Delta x\approx$ 2, 3, 8.5 & 15
mm for $r=$ 1.3, 5, 7.7 and 16.6 mm respectively. We have seen that the
corresponding component of vorticity $\omega_{y}$ decreases due to the
compression applied by the contraction in the $y-$direction. This compression
manifests in sudden change in $R_{\Gamma_{y}}$ from positive to negative,
which could heuristically be associated with buckling of vortices in the
$y-$direction.

At the maximum straining location at $x=$ 130 mm, the $R_{\Gamma_{x}}$
correlation exhibits a larger correlation length, with the first crossover at
a larger $\Delta x$ than for the other two components. This shows that the
straining causes an increase in the correlation length which represents
existence of long coherent structures with majority of them aligned in the
streamwise direction [22]. However, strain appears to have little influence on
the other correlations $R_{\Gamma_{y}}$ and $R_{\Gamma_{z}}$, which are
similar for $x=8$ and $x=130$ mm.

We define a new integral length scale $L_{\Gamma}$ based on $R_{\Gamma}$ as,

$L_{\Gamma}=\int_{0}^{l_{c}}\left|R_{\Gamma}\right|dx,$ (5)

where, $l_{c}$ is the length of the second crossover point from the origin. We
consider up to the second crossover point as there exists a “good” correlation
even after the first crossover though the correlation is negative. To take
this negative-correlation region also into account in characterizing the
integral length scale, we use the absolute value of $R_{\Gamma}$ in Eq. 5. We
present the variation of $L_{\Gamma}$ with loop size, with $x_{1}=$ 10 and 130
mm for both the modes in Figure 7. We see that $L_{\Gamma}$ based on all the
three correlation functions increases with increase in the loop size for both
the modes. This increase with $r$ is fairly linear at the inlet to the
contraction. But there is greater increase seen with $L_{\Gamma_{x}}$ after
the maximum straining location as seen in Figure 7(b). The random mode has
slightly higher values of $L_{\Gamma}$ at the inlet to the contraction. This
can also be seen in the integral scale summarized in Table 1, which is
computed using the correlation of streamwise velocity. This difference could
be due to the difference in the rotation speed of the motors. Due to a longer
duration of structure interaction with the flaps at lower rotation speeds, the
random mode yields higher values of integral scales. The vortical structures
get stretched and tend to align with the $x$ direction. It is to be noted that
the contraction stretches these large vortical structure as they move
downstream. It also tends their principal axis to align in the $x-z$ plane
perpendicular to the $y$ direction, with a higher probability to align with
the $x$ direction. This is the reason for the following order of the integral
scales: $L_{\Gamma_{x}}>L_{\Gamma_{z}}>L_{\Gamma_{y}}$.

###  3.4 Space-time correlation of circulation

|  $x=$10 mm |  $x=$130 mm
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 8: The space-time correlation of circulations: (a) $R_{\tau,\Gamma_{x}}$, (b) $R_{\tau,\Gamma_{y}}$, and (c) $R_{\tau,\Gamma_{z}}$ for the Synchronous grid-rotation mode and loop size of $r/\eta=$ 61.5. The correlation is between the first points $x_{1}=$10 mm (left), $x_{1}=$130 mm (right) and points with $\Delta x=$ 0 (—), 10.2 mm (—), 20.5 mm (—), 30.8 mm (—), 41.1 mm (${--}$), 51.3 mm (${--}$), 61.6 mm (${--}$) and 71.9 mm (${--}$). The vertical lines represent the predicted location of the maximum correlation based on the integrated advection time $\tau_{adv}$ from Eq. (7). |  $x=$10 mm |  $x=$130 mm
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 9: The space-time correlation of circulations: (a)
$R_{\tau,\Gamma_{x}}$, (b) $R_{\tau,\Gamma_{y}}$, and (c)
$R_{\tau,\Gamma_{z}}$ for the Random grid-rotation mode and $r/\eta=$ 51.1.
For legends refer to the caption of figure 8.

We also look at the space-time correlations of the circulation. The space-time
correlation function is defined as

$R_{\tau,\Gamma_{x}}(x_{1},\Delta
x)=\frac{\left\langle\gamma_{x}(x_{1},t)\gamma_{x}(x_{1}+\Delta
x,t+\tau)\right\rangle}{\left\langle\gamma_{x}(x_{1},t)^{2}\right\rangle^{0.5}\left\langle\gamma_{x}(x_{1}+\Delta
x,t+\tau)^{2}\right\rangle^{0.5}},$ (6)

where $\Delta x$ is the space shift and $\tau$ is the time shift. Figure 8
shows the space-time correlation for $x_{1}=$ 10 and 130 mm for mode S1 and
the largest loop with $r/\eta=$ 61.5. The space-time correlation is obtained
by correlating the fluctuating circulations at two spatially separated points
by varying $\tau$ [29],[30]. For smoothness of the correlation function, we
use a time shift that corresponds to ten frames of the high-speed video. With
reference to Figures 8 and 9 we now look at some of the common characteristics
of these space-time correlation functions. For $\Delta x=$0 this simplifies to
the autocorrelation in time, which by definition becomes 1 when $\tau=$ 0 and
decreases to zero at $\tau\approx$ 0.08 s, following which it becomes slightly
negative with further increase in $\tau$. The zero-crossover point of the
correlation function gives a characteristic time scale. On the other hand,
keeping $\tau=0$, $R_{\tau=0,\Gamma_{x}}$ we recreate the space-only
correlation already presented in Figure 6 and encoded on the intersection with
the $y-$axis ($\tau_{adv}=0$) of the curves for various $\Delta x$.

The full space-time correlations, for example for
$R_{\tau,\Gamma_{x}}(x_{1}=10,\Delta x=10.2)$ mm, the correlation grows (from
0.3 when $\tau=$0) with increase in $\tau$ to reach a maximum at $\tau\approx$
0.03 and then decreases, falling to zero at $\tau\approx$ 0.11 s. The
correlation is close to symmetric about this peak. The location of this peak
can be predicted by the Taylor hypothesis of frozen turbulence. We use an
analogy to a passive fluid element transported by the mean flow to understand
this. The spatial position of the fluid element which is initially at $x_{1}$
at time $\tau=$ 0 can be predicted knowing the mean advection velocity
$\left\langle U\right\rangle(x)$ of the transporting flow. The element is
advected through a distance $dx=\left\langle U\right\rangle d\tau$ in a time
period $d\tau$. In line with this element transport, the circulation
fluctuations are also transported by the mean flow. The mean velocity profile
$\left\langle U\right\rangle(x)$ along the centerline is precisely known from
our measurements and hence the advection time can be determined. The advection
time $\tau_{adv}$ for a fluid element to be transported from point $x_{1}$ to
$x_{1}+\Delta x$ is given by,

$\tau_{adv}=\int_{x_{1}}^{x_{1}+\Delta x}\frac{dx}{\left\langle
U\right\rangle(x)}.$ (7)

The advection times for the chosen, space-shifted points, with $\Delta x=$10.2
to 71.9 mm are indicated by the vertical lines in the figures. These lines
coincide exactly with the advection times at the peaks in the corresponding
correlations $R_{\tau,\Gamma_{i}}$. The decrease in the peak values is
expected from the dynamical evolution of the turbulent rotational structures
during the transport by the mean flow. The measurement errors between
sequences of time-volumes also contributes to the reduced correlation peaks.

Figures 8 and 9 show the space-time correlation results for the Synchronous
and Random grid-rotation modes, respectively. The left columns show results
close to the inlet of the contraction where $x_{1}=10$ mm and right column
towards its exit at $x_{1}=130$ mm. For the latter location, the peaks are
spaced by much shorter time intervals as expected due to increase in the mean
advection velocities owing to the reduced cross-sectional area. The most
striking non-trivial difference between the three directional circulations is
manifest in the width of the correlation curves, particularly near the end of
the contraction. The width of the $R_{\tau,\Gamma_{x}}$ curves, measured
between the first and second crossing, is much larger than for the
corresponding $R_{\tau,\Gamma_{y}}$ and $R_{\tau,\Gamma_{z}}$. To minimize
effect of noise we measure the width of the curves around the peak, from the
0.1 correlation levels. Based on the time between first and second crossings
of 0.1, we can define a timescale $T_{i}$, indicating how far in $\Delta\tau$
the rotational structures are correlated for $\Gamma_{x}$, $\Gamma_{y}$ &
$\Gamma_{z}$. These curves show $T[x,y,z]=[0.11,0.046,0.046]$ s for the
Synchronous mode and $T[x,y,z]=[0.18,0.044,0.070]$ s for the random mode. The
correlation time-duration for $\Gamma_{x}$ is therefore 2-3 times longer than
in the other cross-stream directions, strongly supporting the presence of
enhanced streamwise coherent vortices.

|  $x=$ 8 mm |  $x=$ 190 mm
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 10: The probability density function (PDF) of circulation $\Gamma_{x}$
(a), $\Gamma_{y}$ (b) and $\Gamma_{z}$ (c) measured with synchronous grid-
oscillation mode at inlet $x=$ 8 mm (left) and outlet $x=$ 190 mm (right). The
circulation is computed for square loops of size $r\sim$ 4$\eta$ (near the
dissipative scale), 25$\eta$ (the inertial scale) and 50$\eta$ (the large
scale). The PDF is computed for circulation after subtracting its mean value
using 100 bins. The bin values are normalized by the corresponding r.m.s
value. For clarity, two of the curves are shifted vertically by a relative
factor of 2.5. The Gaussian distribution is shown by the continuous dashed
curves.

###  3.5 Probability density function of circulation

The PDFs of circulation computed at two different streamwise locations, $x=$ 8
mm, downstream of $SOC$ and $x=$ 190 mm downstream of $EOC$ are shown in
Figure 10. The circulation is computed using square loops around the
centerline of the measurement volume. The size of the loop is varied from 3
$-$ 27 nodes which corresponds to 1.3 mm to 17 mm. The maximum size of the
loop is limited by the depth of the measurement volume which is approximately
18-19 mm in our case. Circulation computed over nine square loops around the
chosen streamwise location is used in each time-step for the PDF calculations.
The circulation data obtained from 4$-$6 runs conducted on the same day and
for the same grid-rotation protocol are combined together to increase the tail
resolution of the PDFs. With each run having about 50,000 time steps, the PDFs
are generated with $\sim$ 2 million square loops.

Figure 10 shows PDFs obtained with the synchronous mode, for loops of three
different sizes, i.e. $r\sim$ 4$\eta$, near the dissipative scale; 25$\eta$
which is the typical inertial scale (based on the values of $\lambda$ in Table
1); and 50$\eta$ which corresponds to the large scale. The PDFs for the
different loops sizes have been shifted to minimize clutter, in such a manner
that the peak values of two of them are shifted up vertically by the same
distance, i.e. by a factor of 2.5. The PDFs with the random mode are not shown
herein, but are qualitatively similar to the PDFs obtained with the
synchronous mode.

The circulation PDFs, shown in Figure 10, all show Gaussian behavior for the
largest loop size, but deviate from the Gaussian, having wider tails, as the
loop is reduced in size to approach the dissipation scales. The widest tails
are measured for $\Gamma_{z}$, as characterized below by the flatness factor.
The PDFs of circulation on all three planes exhibit this behavior, though the
degree of deviation from Gaussianity are different. This transition from non-
Gaussian to Gaussian with $r$ is more prominent for $\Gamma_{z}$, see Figure
10 (c). The deviation from the Gaussian implies intermittency of the vortex
structures [4],[14]. This behavior of the PDF has been reported in PIV
measurements in the cylinder wake [3], in buoyancy-driven turbulence [14] and
DNS of isotropic turbulence [4],[19]. Cao et al. [4] fitted the stretched
tails to exponential forms with exponents varying with the scales. Iyer et al.
[19] showed that the PDFs collapse into a single curve exhibiting a self-
similarity when the dimensions of the rectangular loops lie in the inertial
range. We look at the time traces of the instantaneous circulations normalized
by their respective r.m.s values, plotted in Figure D1. Let us look at Figure
D1(a) which shows the time variation for the synchronous mode at $x=$ 8 mm,
with left and right panel corresponding to $r\sim$ 4$\eta$ and 50$\eta$
respectively, for one realization of 50,000 volumes. Note that there is
frequent occurrence of extreme values of $\Gamma_{z}$ (with
$|\Gamma_{z}/\Gamma_{z,rms}|>$3). Though such extreme values are seen with
$\Gamma_{x}$ and $\Gamma_{y}$, they are relatively less frequent with
$\Gamma_{x}$ and even fewer for $\Gamma_{y}$. This indicates that $\Gamma_{z}$
is more intermittent compared to the other two components which manifests as
widened PDF tails, with the biggest departure from Gaussian. With all
circulation components, we see the rate of variations reduces with increase in
$r$. The frequent occurrence of extreme values with $\Gamma_{z}$ with $r\sim$
4$\eta$ do not exist with $r\sim$ 50$\eta$, which explains its prominent
transition from the non-Gaussian to Gaussian behavior in the PDF. A similar
observation with prominent transition in $\Gamma_{z}$ can be made at $x=$ 190
mm shown in Figure D1(b).

|  $x=$ 130 mm |  $x=$ 190 mm
---|---|---
(a) |  |
(b) |  |
Figure 11: The flatness for probability density function (PDF) of circulation
$\Gamma_{x}$ ($\circ$), $\Gamma_{y}$ ($\square$) and $\Gamma_{z}$
($\triangle$) measured at $x=$ 130 mm (left) and 190 mm (right) with the
synchronous grid-oscillation mode (a) and the random mode (b). The flatness is
computed as the fourth moment of the distribution normalized the r.m.s value,
$\left|\left\langle\Gamma^{4}\right\rangle\right|/\left|\left\langle\Gamma^{2}\right\rangle\right|^{2}$.
The Gaussian value is indicated by the horizontal dashed line.

To quantify the deviation from Gaussian, we plot the flatness of the obtained
distribution for the entire range of scales for both the grid-rotation modes
in Figure 11. The values are plotted at two locations after straining: $x=$
130 mm which is slightly downstream of the maximum strain location, and at
$x=$ 190 mm slightly after the exit of the contraction. The Gaussian flatness
equals 3 and is indicated in the figure by dashed lines. With $\Gamma_{z}$, we
see the expected behavior of flatness tending to the Gaussian value as the
loop size increases. Due to its higher intermittency, a notable decrease in
its flatness is seen with increase in $r$ and this continues till the largest
loop. For $r\sim$ 50$\eta$ the flatness is $\approx$ 3.2 & $\approx$ 3.4 with
the synchronous & random modes respectively, as seen in Figure 11. However,
with circulations in the streamwise and transverse planes ($\Gamma_{x}$ and
$\Gamma_{y}$), there is marginal or no decrease in the flatness values with
change in the loop sizes. This indicates weak intermittency in $\Gamma_{x}$
and $\Gamma_{y}$ and is inline with their time signals in Figure D1. The
tendency of flatness of circulation towards constancy in the IR has been
observed by Iyer et al. [19] with increase in $Re_{\lambda}$ compared to the
flatness of the velocity increments. Based on this they conclude that $\Gamma$
is weakly intermittent and the velocity increments are highly intermittent.
This could be due to the stronger effect of straining on circulation in these
two planes. We have seen that the vorticity component in the streamwise and
transverse directions undergo maximum variation due to straining in the same
contraction [22]. Higher values of flatness were observed close to the $SOC$,
which indicates higher intermittency, which is due to the influence of the
grid protocol and slightly higher $Re_{\lambda}$. A greater tendency for the
PDFs towards Gaussian at lower $Re_{\lambda}$ has been reported by Cao et al.
([4]).

| Synchronous Mode | Random Mode
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 12: The moments of circulation-PDF,
$\left\langle\left|\Gamma\right|^{n}\right\rangle$ and its scaling computed at
$x=$ 190 mm with mode S1 (left) and the random mode (right). (a) The moments
of $\Gamma_{z}$ of order $n=$ 2 ($\circ$), 3 ($+$), 4 ($\bigtriangleup$), 5
($\times$), 6 ($\diamond$), 7 ($\ast$) and 8 ($\bigtriangledown$). The black
line represents the best power-law fit of the form
$\left\langle\left|\Gamma\right|^{n}\right\rangle\sim\left(r/\eta\right)^{\lambda_{n}}$
for the data in the intermediate scales. (b) The local slopes of the data
shown in (a) which is computed as
$\Delta[log\left\langle\left|\Gamma\right|^{n}\right\rangle]/\Delta[log\left(r/\eta\right)]$.
The local slopes are shown for the moments of order $n=$ 2$-$5\. The horizonal
dashed lines with the symbol on it represent the corresponding K41 value. (c)
The values of power-law exponent for moments of order $n=$ 2$-$8\. The solid
line indicates the K41 scaling, $\lambda_{n}=4n/3$. The dotted lines indicate
the fits proposed by ISY19 (Iyer et al. (2019) [19]) i.e., for $n<$ 3,
${\lambda_{n}}=1.367n$ and $n>$ 3, ${\lambda_{n}}=1.1n+(3-D)$, where $D$ is
2.2.

###  3.6 Moments of the circulation

We look at the scaling properties of the circulation moments, which are
computed as (8).

$\left\langle\left|\Gamma\right|^{n}\right\rangle=\int\left|\Gamma\right|^{n}f(v)dv.$
(8)

Here, $f$ is the circulation PDF, and the absolute value of the circulation is
used to include the odd-order moments in the scaling. Iyer et al. [19] suggest
the scaling improves if the absolute value is used and we follow their
example. Keep in mind that our geometry retains symmetries, along the
centerline, so the mean values of all components should be zero. These moments
are also referred to as the circulation structure functions [14], definition
being in line with the velocity structure functions. According to the
Kolmogorov theory (K41) for homogeneous isotropic turbulence, the circulation
structure functions have a power-law scaling, $\sim(r/\eta)^{\lambda_{n}}$,
where the exponent $\lambda_{n}=$ 4$n/$3 [13]. The velocity structure function
has a lower value for its exponent which is $\zeta_{n}=$ $n/$3\. However,
based on the simulations at high Re${}_{\lambda}\sim$ 1300, Iyer et al. [19]
proposed two fits for these exponents based on the value of $n$. The proposed
fits are included in the caption of Figure 12. The moments of circulation-PDF
is plotted in Figure 12(a) for order $n=$ 2 to 8 and for both the grid-
rotation modes. We restrict to only order 8, considering the low accuracy of
the technique in prediction of higher order statistics. The power-law fit that
approximates the data in the intermediate scales ($r/\eta>$ 10) are also
shown. The corresponding local slopes of the data shown in Figure 12(a) is
plotted in Figure 12(b) assuming a linear variation between adjacent points.
We see that the local slopes seen with $n=$ 2, remains fairly constant over a
range of scales with $r/\eta$ above 10. Hence we chose this range of scales to
obtain the power-law fits. The exponent of the fits for both the modes and
circulation in the three planes at $x=$ 190 mm are plotted in Figure 12(c). If
the PDF has a Gaussian distribution, the moments should follow the K41
scaling. It is seen that in all cases, the exponents have sub-Kolmogorov
scaling which has been seen with experiments and simulations reported in the
literature [3, 4, 19]. The deviation from K41 increases with increase in $n$.
It can be seen from figures 12(c) that of the three circulations the
$\Gamma_{z}$ most closely follows the K41 (the solid lines in the figure),
followed by the streamwise $\Gamma_{x}$ and the transverse $\Gamma_{y}$, which
fall below the prediction. The exponents of the fits near the contraction
inlet are shown in Figure D2. Here, $\Gamma_{z}$ follows K41, however there is
a change in the deviation from K41 with the other components. Larger departure
from K41 is seen for $\Gamma_{x}$. Zhou et al. [14] note that such deviation
from K41 could be due to intermittency or the anisotropy of the coherent
structures. The moments of the spanwise $\Gamma_{z}$ do not undergo much
change between inlet and outlet of the contraction. On the other hand, the
compression of the vortical structures manifests as the increase in the
deviation of the moments of $\Gamma_{y}$. This is inline with the flatness
shown by the streamwise and transverse circulations, which is attributed to
the intermittency. As pointed out previously, in the large scale, the vortex
filaments experience extension and compression along these directions
respectively. The streamwise $\Gamma_{x}$ and the transverse $\Gamma_{y}$
undergo the straining effects. Of the two grid-rotation modes, the random mode
shows lesser deviations from K41 as compared to the mode S1. However for both
modes, the exponents are closer to the scaling given by ISY19 (Iyer et al.
(2019) [19]), which are based on the DNS data of homogeneous isotropic
turbulence.

|  $\Delta x=$ 10 mm |  $\Delta x=$ 21 mm
---|---|---
Condition |  $R_{\Gamma_{x}}$ |  $R_{\Gamma_{y}}$ |  $R_{\Gamma_{z}}$ |  $R_{\Gamma_{x}}$ |  $R_{\Gamma_{y}}$ |  $R_{\Gamma_{z}}$
No deformation |  0.53 |  0.25 |  0.35 |  0.27 |  $-$0.01 |  $-$0.06
With deformation |  0.46 |  0.32 |  0.39 |  0.22 |  $-$0.03 |  $-$0.05
|

Table 2: Comparison of the correlation values $R_{\Gamma_{x}}$,
$R_{\Gamma_{y}}$ and $R_{\Gamma_{z}}$ obtained with loops with and without
considering deformations from the mean strain, calculated for the synchronous
grid-rotation mode. The correlation is computed with loops which are initially
a 16.1-mm square at $x_{1}=$ 130 mm. The variation of the correlation with
$\Delta x$ for loops without deformation is shown in Figure 6. The loop after
deformation during the respective space shifts is indicated by red dashed
lines.

###  3.7 Circulation with loop deformation

The spatial correlations of the circulations were computed using a fixed
square loop shape and area inside the contraction. However, the material
elements and loops get deformed by the mean strain as they are advected
through the contraction. To estimate the effect of this on the correlation
functions, we compare circulation statistics with an without deforming the
loop-shape, while keeping their area constant. Table 2 sketches the
deformation of an initially 16.1-mm square-loop at $x_{1}=$ 130 mm, which gets
stretched by 2 mm in the $x$ direction as it is advected by 10 mm downstream;
and by 3 mm as it is advected by 21 mm, with a corresponding squeezing in the
$y-$direction and no change in its $z-$direction. Table 2 lists the resulting
changes in the correlation values at $x=10$ mm of ($-$0.07, 0.07, 0.04) for
($R_{\Gamma_{x}}$, $R_{\Gamma_{y}}$, $R_{\Gamma_{z}}$) respectively. Further
downstream, for $\Delta x=$ 21 mm, the corresponding differences were
($-$0.05, $-$0.02, 0.01), where $R_{\Gamma_{x}}$ remains strong, while
$R_{\Gamma_{y}}$ and $R_{\Gamma_{z}}$ have values close to zero. We conclude
that the rather modest deformation of the loops does not alter our
conclusions.

##  4 Discussion and conclusions

(a) | (b)
---|---
|
Figure 13: The probability density function (PDF) of the relative circulation
strengths at (a) $x=$ 8 mm which is near the start of contraction, and (b)
$x=$ 190 mm which is after the exit. The PDFs are measured with the random
mode for loop size $r\sim$ 50$\eta$ using 50 bins. At the inlet all the three
circulation components are about evenly distributed, while at the exit the
circulation is strongest in the streamwise stretching direction $\Gamma_{x}$
and weakest in the transverse compression direction $\Gamma_{y}$.

Herein we have experimentally determined the statistical evolution of the
three components of circulation in turbulent flow that is convected through a
two-dimensional smooth contraction. Turbulent fluctuations were injected into
the flow using an active grid operated in either synchronous or random
rotation modes, to reach $Re_{\lambda}\approx$ 220\. We have used the latest
3-D LPT with the STB algorithm to measure the time-resolved three-dimensional
velocity fields in volume-slices inside the contraction. This enabled us to
calculate the simultaneous circulations in loops contained in three mutually
perpendicular planes, casting these circulations in terms of a vector. This
allow us to pinpoint the effects of the mean strain on the circulation
components and their effective orientation. Figure 5 reveals this orientation
by comparing the PDF of the individual directional components $\Gamma_{x}$,
$\Gamma_{y}$ and $\Gamma_{z}$ normalized by the instantaneous absolute value
$\left|\Gamma\right|$ of the vector. While the $\Gamma_{x}$ circulation around
a fixed-size loop, will clearly increase from the streamwise stretching and
enhancement of vorticity within the loop, the circulation will also increase
by the enhanced streamwise alignment of the vortices, see Mugundhan et al.
(2020). We interpret the orientation of the circulation vector as indicating
the corresponding orientation of large-scale coherent vortices. This is
highlighted in Figure 13, which compares the three PDFs at the inlet in (a)
and the outlet of the contraction in (b). At the inlet all the three
circulation components are evenly distributed, with a nearly flat
distribution, while at the exit the component PDFs vary greatly. They show
that the straining of the contraction causes orientation with $\Gamma_{x}$ to
be the strongest (by the largest probability at
$\Gamma_{x}/\left|\Gamma\right|=\pm 1$), while the weakest orientation is in
the contraction direction, i.e. $\Gamma_{y}/\left|\Gamma\right|$ peaking
around 0.

This effective orientation can also be seen with the behavior of the
streamwise integral length scales $L_{\Gamma}$ deduced from the streamwise
spatial correlation of the circulation components. The straining causes a
strong increase in $L_{\Gamma_{x}}$ while the other components are mostly
unchanged, as shown in Figure 7. We also see that the random grid-oscillation
mode has larger integral scales than the synchronous mode. This is due to the
lower grid rotation speed used in the random mode, 3.5 vs 4.0 rps. Low grid-
rotation speeds enables longer flap-coherent structure interaction time and
hence results in larger integral scales. Similar observations were made by
Larssen & Devenport [27]. The length scales based on the two-point velocity
correlations in active-grid generated turbulence in wind tunnel have been
reported [27, 28] but are based on point measurements using Taylor’s
hypothesis of frozen turbulence. Larssen & Devenport [27] used a macro length
scale $=k^{3/2}/\epsilon$ as a substitute for integral length scale. They
observed larger macroscales at lower rotation speeds of the active grid.

The circulation PDFs all show Gaussian behavior for the largest loop sizes
(Figure 10), but deviate from the Gaussian, with wider tails, as the loop is
reduced in size towards the dissipative scales. The widest tails are measured
for $\Gamma_{z}$. This deviation is characterized by the flatness factor
(Figure 11), which shows the clearest transition away from Gaussian for
$\Gamma_{z}$ evaluated on small loops. On the other hand, the flatness of
$\Gamma_{x}$ and $\Gamma_{y}$ remain fairly near the Gaussian value between
$3-3.4$. This may indicate that $\Gamma_{x}$ and $\Gamma_{y}$ are weakly
intermittent compared to the spanwise $\Gamma_{z}$. The power-law scaling of
the moments of $\Gamma$ shows sub-Kolmogorov scaling for all three components.
Large deviations are seen in the compressed direction of $\Gamma_{y}$ at the
exit of the contraction. The PDF satisfies approximately the area rule with
only 10% deviations in the rms values, which we discuss in Appendix F. A
larger sample would be needed to verify this rule, which is beyond the current
experiments.

In closing, while the commonly used statistical characterizations of
circulation, such as pdf and scaling exponents, have proven powerful to
characterize the dynamics of homogenous turbulence, we find that for
turbulence exposed to large-scale strain it is more effective to measure the
simultaneous circulations in three perpendicular planes. This allows us to
construct a circulation vector and pinpoint how the strain enhances and
orients coherent vortical structures, by instantaneously following the
relative strength of the components, as is implied by the PDFs in Figure 13.
Keep in mind that this has only become possible experimentally with the
recently available volumetric measurements of velocity, employed herein.

## Acknowledgements

This study was supported by King Abdullah University of Science and Technology
(KAUST) under grant BAS/1352-01-01.

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##  Appendix A Relative strength of circulation components

|  $r\sim$ 28$\eta$ |  $r\sim$ 60$\eta$
---|---|---
(a) |  |
(b) |  |
(c) |  |
Figure 14: The asymmetry of the circulations for the synchronous grid-
oscillation mode. The streamwise variation of probability density function
(PDF) of relative circulation strengths $\Gamma_{x}/\left|\Gamma\right|$ (a),
$\Gamma_{y}/\left|\Gamma\right|$ (b) and $\Gamma_{z}/\left|\Gamma\right|$ (c)
measured with grid-rotation mode S1 for loop sizes $r\sim$ 28$\eta$ (left) and
$r\sim$ 60$\eta$ (right). Refer to caption of Figure 5 for more details. Note
the much larger asymmetry here for mode S1 than the random mode in Figure 5.

##  Appendix B Homogeneity of the mean and r.m.s vorticity field

| Synchronous Mode | Random Mode
---|---|---
$\left\langle\omega_{x}\right\rangle$ |  |
$\left\langle\omega_{y}\right\rangle$ |  |
$\left\langle\omega_{z}\right\rangle$ |  |
Figure 15: Contour plots of the time-averaged vorticity field ($s^{-1}$) in the mid-$z$ plane of the measurement volume $P1$. The top, middle and bottom rows correspond to the streamwise component $\left\langle\omega_{x}\right\rangle$, the transverse component $\left\langle\omega_{y}\right\rangle$ and the spanwise component $\left\langle\omega_{z}\right\rangle$ respectively. The $X-Y$ co-ordinates shown in the figure are local to the measurement region as used in the calibration. The origin (0,0) here is the local coordinates and it corresponds to (42,0) in the general coordinate system $x-y$ shown in Figure 1. The range of the vertical scale $X=$ [44,$-$56] thereby corresponds to $x=$ [$-$2,99] in the general coordinate system. The time averaging is done over 50,000 images captured at 1300 frames-per-second. The overall magnitude values on the centerline are close to zero as expected from $x-$symmetry. However, more inhomogeneity is observed with the synchronous mode. | Synchronous Mode | Random Mode
---|---|---
$\omega_{x,rms}$ |  |
$\omega_{y,rms}$ |  |
$\omega_{z,rms}$ |  |
Figure 16: Contour plots of the r.m.s vorticity field ($s^{-1}$) in the
mid-$z$ plane of the measurement region $P1$. The top, middle and bottom rows
correspond to the streamwise component $\omega_{x,rms}$, the transverse
component $\omega_{y,rms}$ and the spanwise component $\omega_{z,rms}$
respectively. Refer to Figure 15 for details on the coordinates. The
synchronous mode shows much larger asymmetries in $\omega_{x,rms}$ and
$\omega_{y,rms}$, than the random mode.

##  Appendix C Mean velocity field

Synchronous Mode | Random Mode
---|---
|
|
Figure 17: Contours of the time-averaged streamwise velocity field
$\left\langle U\right\rangle$ ($ms^{-1}$) and vector plots of the time-
averaged velocity field $\left\langle(U,V)\right\rangle$ ($ms^{-1}$) in the
mid-$z$ plane of the measurement volumes, P1 (top row) and P2 (bottom row).
The $X-Y$ co-ordinates shown in the figure are local to the measurement
region, as used in the calibration. In P1, the origin (0,0) here is the local
coordinates and it corresponds to (42,0) in the general coordinate system
$x-y$; and in P2 the origin corresponds to (147,0) in the general coordinate
system shown in Figure 1. Thus, the range of the vertical scale in P1, $X=$
[44,$-$56] corresponds to $x=$ [$-$2,99]; and in P2 $X=$ [48,$-$58]
corresponds to $x=$ [100,206]. Only a few of the vectors are shown for
clarity. For P1, vectors are skipped for every (12,12) grid points and in P2
it is skipped for every (6,12) grid points.

##  Appendix D Intermittency of Circulation & Scaling Exponents

$r\sim$ 4$\eta$ |  $r\sim$ 50$\eta$
---|---
(a)

(b)
---

Figure 18: The time traces of instantaneous circulation normalized by the respective r.m.s, $\Gamma_{i}/\Gamma_{i,rms}$ computed for square loops around the centerline of the contraction of size $r\sim$ 4$\eta$ (left) and $r\sim$ 50$\eta$ (right) for the synchronous mode at (a) $x=$ 8 mm, and (b) $x=$ 190 mm. The $x-$axis represents the frame number and each frame is spaced by $\approx$ 0.8 ms. (a) | (b)
---|---
|
Figure 19: The power-law exponent for moments of circulation-PDF,
$\left\langle\left|\Gamma\right|^{n}\right\rangle$, at $x=$ 8 mm with
synchronous (a) and the random mode (b). Refer to caption of Figure 12 for
details.

##  Appendix E Uncertainty and Error quantification

(a) | (b)
---|---
|
(c)
Figure 20: The histogram of uncertainty in the particle velocities $u$ (a),
$v$ (b) and $w$ (c) predicted using the particle trajectories in the
measurement region P1. The uncertainty is obtained using $\sim$109 particle
samples over 20,000 volumes.

We present the uncertainty calculated using the DaVis 10.2 program from
LaVision. The uncertainty is estimated for the particle tracks and the
velocities. The estimation is based on the method by Janke and Michaelis [31],
wherein they use linear regression analysis to calculate confidence bands of
the polynomial fits of the particle trajectories to estimate the uncertainties
in the particle positions and velocity. The histogram of uncertainty values
from using $\sim 10^{9}$ particles from about 20,000 volumes is shown in
Figure 20. The 1-sigma values of uncertainties in the measurement regions, as
a percentage of the mean inlet velocity are:

(1) P1: velocity $u$, $v$ & $w$ are 1.4%, 1.5% & 2.5% respectively.
(2) P2: velocity $u$, $v$ & $w$ are 1.6%, 1.8% & 4.0% respectively.

As the flow is accelerating, the percentage is based on the mean velocity at
the inlet to the contraction, which gives a conservative local effect.

How does this compare to other studies in the literature? Sciacchitano et al.
[32] compared the accuracy of the various Lagrangian Particle Tracking
algorithms by various research groups, which included the LaVision algorithm
that we use in the current study. The comparisons were done using synthetic
experimental data of a turbulent wall-bounded flow in the wake of a cylinder.
The algorithms accuracy in predicting velocities were in the range 0.3 $-$
2.5% for particle concentration varying between 0.005 and 0.016 particles-per-
pixel (ppp), which is in line with what we see herein.

The actual uncertainties in the particle position are 0.08, 0.1 and 0.2 pixel
in the $x$, $y$ and $z$ directions which is also inline with the Lagrangian
particle tracking measurements using shake-the-box algorithm in the
literature. Rowin and Ghaemi [33] measured near-wall turbulent flow over a
superhydrophobic surface using 3-D LPT. They used DaVis 8.4 to implement the
shake-the-box algorithm and report in-plane & out-of-plane position accuracy
of 0.1 & 0.2 pixel respectively. Schröder and Schanz [34] in their recent
review of the latest 3-D Lagrangian particle tracking methods, report that the
position accuracy of the particle in LPT is typically around 0.1 pixel.

We quantify the uncertainty in the circulation based on the uncertainty in the
velocities by performing Monte Carlo simulations. Circulation is computed for
a loop with an error imposed on the measured velocities using random numbers
as $\Gamma_{i,e}$. The random number imposed on each velocity component
follows a normal distribution with zero mean and the corresponding standard
deviation reported above. The error in the circulation is computed as
$\Gamma_{i,e}-\Gamma_{i}$, where $\Gamma_{i}$ is the actual circulation
without the imposed error. Considering the 75% overlap of adjacent grid
points, relative to the interpolation from particle tracks, the random error
in every fourth velocity vector is assumed to be independent and thus
conservatively the random number is kept the same over 4 consecutive grid
points. Therefore, for a contour of length $N=$16 $dx$, we estimate the error
$\epsilon_{\Gamma}=\epsilon_{u}/\sqrt{4}$. These calculations are performed
over 1000 samples to obtain the RMS of the errors. The RMS thus obtained for
an $r=$ 10 mm square loop, for $\Gamma_{x}$, $\Gamma_{y}$ and $\Gamma_{z}$ are
about 2.1%, 2.2% and 1.1% respectively. In line with the velocity error
quantification, these percentages are based on mean velocity at the inlet to
the contraction and the loop size $r$. The reduced error from $u$ to $\Gamma$
is consistent with Sciacchitano and Wieneke [35] who discuss the uncertainty
of vorticity for PIV measurements, suggesting that the uncertainty in
vorticity can be reduced by computing it using circulation around a small
loop, rather than taking velocity derivatives.

(a) | (b)
---|---
|
Figure 21: (a) Mean velocity and RMS vorticity with error bars. (b) RMS
velocity and RMS circulation in region P2, for the synchronous mode. Error
bars correspond to the RMS variation between the six experimental runs. For
clarity, bands have been shown for every third point in $x$.

The typical measurement convergence error bars associated with the mean and
RMS quantities is shown in Figure 21. The convergence error computations are
made using the six independent realizations for the synchronous mode, inside
the contraction in region P2. Error bands shown in the figure correspond to
the RMS of the measurements in six different experimental runs.

The average values for RMS as a percentage of the corresponding mean values
are:
(1) Mean velocity $\left\langle U\right\rangle$ is 0.2%;
(2) RMS velocities $u_{rms}$, $v_{rms}$, $w_{rms}$ are 2.0%, 1.6%, and 1.1%;
and
(3) RMS vorticity components $\omega_{x,rms}$, $\omega_{y,rms}$,
$\omega_{z,rms}$ are 1.4%, 1.3%, and 1.2% respectively.

For the circulations computed for an $r=10$ mm square loop, these values are:
(4) RMS circulation $\Gamma_{x,rms}$, $\Gamma_{y,rms}$, $\Gamma_{z,rms}$ are
4.5%, 2.0%, and 3.2% respectively.

##  Appendix F Testing the Area Rule

Earlier numerical and theoretical studies have focused on the so-called area
rule of the circulation [2, 19]. Our volumetric data can potentially allow us
to test this experimentally. For this purpose we have computed the PDF of the
streamwise circulation $\Gamma_{x}$ with both square and rectangular loops,
which have the same area. For example loops with outer dimensions of
$5.1\times 5.1$ mm2 vs $2.6\times 10.2$ mm2. The loops are taken around the
center of the measurement region, just inside the start of the contraction, at
$x=$ 8 mm. The circulation PDFs obtained for loops of area $A\approx$ 25 and
100 mm2, but three different aspect ratios, are shown in Figure 22. The
normalized PDF for the different loop shapes matches to a great extent, while
having some deviations at the tails, which supports the area rule. However,
the deviations of the tails are due to the limitation in the sampling volume,
which in our case are limited by the memory of the cameras. The larger loop
area shows better match of the PDFs, as seen in Figure 22(b). Similar match of
PDFs is seen after the exit of the contraction at $x=$ 190 mm. On the other
hand, the r.m.s values of $\Gamma_{x}$ varied between 9%$-$20% for the 25 mm2
loops, and 2%$-$7% for 100 mm2 loops. A larger sample is thus needed to verify
the area rule to cover extreme probabilities at the tails, which is beyond the
current experiments.

(a) | (b)
---|---
|
Figure 22: Testing the area rule for $\Gamma_{x}$. Comparison of circulation
PDFs for square and rectangular loops having the same enclosed area. The area
of the loops are approximately equal to (a) 25 mm2 and (b) 100 mm2. The bin
values are normalized by the corresponding r.m.s value. The Gaussian
distribution is shown by the continuous dashed curves. Data of $\Gamma_{x}$
with synchronous grid-oscillation mode, at inlet $x=$ 8 mm.