# Interaction of Vortex Ring with Perforated V-Wall

Siddhant Jain  Saini Jatin Rao  Saptarshi Basu<EMAIL_ADDRESS>Department of
Mechanical Engineering, Indian Institute of Science, Bengaluru, India

###### Abstract

Experiments are performed to investigate the interaction of a vortex ring
(Reynolds number based on circulation (ReΓ = 11500) with perforated surface
(open area ratio, $\phi_{1}=0.24$ and $\phi_{2}=0.44$) with different included
angles ($\theta$ = 60° - 180° ). The phenomenon is characterized using
techniques like Planer Laser-Induced Fluorescence (PLIF) imaging and Particle
Image Velocimetry (PIV). Lagrangian analysis using finite-time Lyapunov
exponents (FTLE) and $\Gamma_{2}$ vortex identification methods are utilised
to understand the flow physics. Early observations reveal the growth of
induced mushroom structures through the holes as a consequence of placing the
perforated surface in the path of the vortex ring. These structures along with
Kelvin-Helmholtz (K-H) instability imparts the initial instability to the
emerging jets. We discern a sequential emergence of the vortex ring in the
form of jets at lower $\theta$ value that diminishes at higher values. Except
for $\theta$ = 150° cases, where the flow from the two halves starts to talk
resulting in a divergence in the circulation ratio, a reformed vortex ring is
formed for all cases in the far downstream. A detailed discussion on the
downstream vorticity dynamics has been been provided using vorticity contours,
time-series variation of circulation and the FTLE fields. By varying the value
of $\theta$, we present a more generalised study of vortex ring interacting
with perforated surfaces that finds application in multiple domains including
flow control, manipulation and vortical cleaning.

##  1 Introduction

Vortex rings are formed as a result of an impulsive motion of fluid mass
leading to the rolling up of shear layers at the corner of the pipe/hole used
to generate it. They are fluidic structure that are of industrial as well as
fundamental importance. In nature, they can be observed in smoke rings,
jellyfish locomotion (Dabiri et al., 2005), wakes of flying birds (Wolf et
al., 2013), mushroom clouds during volcanic eruptions and explosion (Lim &
Nickels, 1995), micro-burst (Lundgren et al., 1992), cardiac relaxation of
human heart (Per M. et al., 2016), and more. Application wise, they can be
used for flow control purposes (Amitay et al., 2001; You & Moin, 2008), heat
transfer application of jets (Liu & Sullivan, 1996; Hadziabdic & Hanjalic,
2008), particle shaping technique (An et al., 2016), and cleaning purposes
(Jain et al., 2023) to name a few.

From a long time, physicists have been intrigued in understanding the
fundamental behaviour of vortex ring and the interplay of its origin and decay
with the surrounding. Works by Saffman (1970); Maxworthy (1972, 1977); Didden
(1979); Glezer (1988); Glezer & Coles (1990) includes investigations on the
fundamental aspects of vortex ring like the translation velocity, laminar and
turbulent natures, their stability and formation. More recent works by Gharib
et al. (1998); Mohseni & Gharib (1998); Shusser & Gharib (2000); Gan et al.
(2012); Krieg & Mohseni (2013); Xiang et al. (2017) shows the existence of a
universal formation number, discusses formation time scales, drag forces,
modelling of translation properties and energy of the vortex ring.

Of interest to the present work is how vortex rings interact with other bodies
like vortex ring itself, solid or porous surfaces and fluidic interfaces. Lim
& Nickels (1992); Cheng et al. (2018) reported the collision of a vortex ring
with another vortex ring giving rise to vortex reconnection and multiple small
sized vortex rings. The interaction of vortex ring with a flat wall normal and
inclined to the translation axis of vortex ring has been reported (Walker et
al., 1987; Lim, 1989; Orlandi, 1990; Chu et al., 1993; Couch & Krueger, 2011;
New et al., 2016). Interactions with round cylinders (Naitoh et al., 1995; New
& Zang, 2017), sphere (ALLEN et al., 2007) ,V-shaped wall (New et al., 2020),
wavy wall (Morris & Williamson, 2020), concave hemispherical cavities (Ahmed &
Erath, 2023), droplet (Sharma et al., 2021), levitated droplet (Sharma et al.,
2022), air-bubbles (Jha & Govardhan, 2015; Biswas & Govardhan, 2023) can be
found in the literature.

Works involving vortex ring interaction with porous surfaces have received
relatively lesser attention. Adhikari & Lim (2009) conducted initial
experiments to demonstrate the interaction of vortex ring having ReΓ from 384
to 2369 with porous surface (${\phi}$ = 62% and 81%) with constant mesh
diameter (0.2mm). They showed similarities of such an interaction with
interaction of vortex ring with non-porous walls. Hrynuk et al. (2012)
reported the formation of a coherent downstream vortex ring only in case of
fine mesh. In coarser mesh, chaotic downstream flow was observed because of
small-scale vortical shedding from the cylindrical structures of the mesh.
Naaktgeboren et al. (2012) proposed a model relating the kinetic energy
dissipation and decrease in hydrodynamic impulse of the vortex ring after
interaction.Cheng et al. (2014) carried out numerical study and reported that
both reducing the ${\phi}$ and increasing the thickness of porous surface has
similar effect i.e., reduction in vorticity transmission, contrary to
increasing the Re of the vortex ring. In another study by Hrynuk et al.
(2018), they argue that the formation of the downstream vortex ring is a
function of vortex shedding at the mesh rather then screen porosity. (Xu et
al., 2018) used FTLE and phase-averaged vorticity fields from the PIV data to
reveal flow evolution in a interaction process. Further, they proposed a
vorticity cancellation mechanism as the reason for the development of
downstream vortex ring. Xu et al. (2019) employed velocity triple
decomposition to understand the interaction at different scales. They found
that surface with largest hole resulted in minimum energy fluctuations calling
it suitable for flow control. Hu & Peterson (2018) experimented on the
interaction of vortex ring with coaxial aperture of different sizes relative
to the size of vortex ring. The concept of vortical cleaning was recently
introduced by Jain et al. (2023) where they utilized the kinetic energy of
vortex ring to clean oil-impregnated porous surface.

Herein, we investigate the problem of a vortex ring interacting with V-shaped
perforated wall. A study with non-permeable V shaped surface was conducted by
New et al. (2020) where they explore the interaction for ${\theta}$ = 30° -
120° at two different ReΓ (2500 and 5000). A detailed discussion on the
formation of secondary and tertiary vortex ring have been provided for
orthogonal and valley plane (check figure 1(a) where xy: valley plane and yz:
orthogonal plane). In the present work, we consider a vortex ring with ReΓ =
11500 and two different open area ratios ${\phi_{1}}$ = 0.24 and ${\phi_{2}}$
= 0.44 for the perforated plate. The included angle, ${\theta}$ is varied from
60° - 180°, the later being a plane perforated surface as shown in figure
1(d). The kind of perforated plate used here has not been reported before that
provides significant resistance to the vortex ring on passing through it. We
use PLIF, PIV, FTLE fields and $\Gamma_{2}$ vortex identification methods to
bring out the flow physics involved during the interaction process. In
particular, we try to answer the following questions: 1) How does a vortex
ring emerge out of inclined perforated walls? 2) How is the downstream flow
modified by the variation in $\theta$? 3) How does the emerging jets behave?
4) How does the ${\phi}$ value change the vortex dynamics in the downstream
region? Further, till date all the studies conducted with
porous/perforated/permeable surfaces have been with $\theta$ = 180° . Whereas,
this works considers a more general case by varying the $\theta$ and therefore
is relevant to all the above mentioned applications.

The paper is organised as follows: Section 2 contains details of the
Experimental setup and methodologies, section 3 discusses qualitative and
quantitative results from the interaction process for both the perforated
surfaces under subsequent subheadings and section 4 presents summary and
conclusions of the present work.

##  2 Experimental Setup

###  2.1 Flow chamber, vortex ring generation and perforated surface

All the experiments were conducted in a water-filled acrylic tank of size 450
mm x 450 mm x 600 mm having wall thickness of 8 mm with the free surface
exposed to atmosphere as shown in figure 1(a). Vortex ring of ReΓ = 11500 was
generated using a solenoid valve connected to a pressurized water chamber. The
solenoid valve, controlled by an Arduino board arrangement was opened for a
specific time (60 ms) to ensure optimal mass ejection from an acrylic pipe of
diameter (D) 12 mm. The ejection pressure was set to 40 psi for all the cases.
From the open end, around 25% of the acrylic pipe was kept underwater to
ensure no effects from the free surface. We use acrylic sheets of 3 mm
thickness (h) to make the perforated surface so as to facilitate optical
accessibility required to conduct PIV and PLIF experiments. Two different
perforated surfaces are considered in the present study having ${\phi}$ values
of 0.24 (${\phi_{1}}$)and 0.44 (${\phi_{2}}$). This was achieved by keeping
the hole diameter (${d_{h}}$) constant at 3 mm and changing the pitch (p) as
shown in figure 2(b). Two different perforated plates were attached at the
centre aligning with the translational axis of the vortex ring to achieve the
desired ${\theta}$ values. A frame with two side stands was designed as shown
in figure 1(a) to hold the two ends of the perforated plate firmly inside the
water. The area covered by the holes on the perforated plate was kept 3 times
larger than the size of the vortex ring. The nomenclature used for each of the
case is in the form of ${\phi_{(1or2)}\theta^{(includedangle)}}$ for example:
${\phi_{1}\theta^{60}}$ refers to the case with open area ratio of 0.24 and
included angle of 60° between the two surfaces. The variation in the value of
the included angle was found to be within $\pm$ 5°.

Figure 1: (a) A 3-dimensional view of the experimental setup (b) Setup for PIV
and PLIF measurements (c) A 2-dimensional view of the experimental setup. The
hatched zone shows qualitatively the area considered for calculation of the
circulation using the PIV data (d) Different configurations of the perforated
surface used in the present study.

###  2.2 PIV and PLIF

The PIV measurements were done using a high-speed dual pulse Nd:YLF laser
(Photonics Inc.,pulse energy of 30 mJ, emission wavelength 527 nm). Both PIV
and PLIF recordings were done at 1000 Hz. A typical setup for both PIV and
PLIF experiments are shown in figure 1(b). For uniform illumination of the
interaction zone, cylindrical lens beam was expanded and converted into a
planar sheet of thickness $\sim$ 1 mm. Neutrally buoyant borosilicate glass
spheres (Signma-Aldrich) of mean diameter ranging from 9-13 ${\mu}$m and
having density of 1100 Kg m-3 were dispersed in water homogeneously and were
illuminated by laser sheet. The seeder particles were also mixed with the
water that was used to generate the vortex ring along with the quiescent fluid
in the tank. The laser entered the acrylic tank from one side (figure 1(b))
that was aligned precisely with the array of holes to acquire accurate flow
field data coming out of the hole. The images were recorded using Photron Mini
UX high-speed camera at a pixel resolution of 1280 pixels x 1024 pixels with a
field of view of $\sim$ 66 mm x 53 mm. Cross-correlation technique with
constant multi-pass interrogation window size of 64 pixels x 64 pixels was
employed to process the displacement vectors. 75% overlap was fixed for all
the cases resulting in vectors at spacing of 0.84 mm. The captured PIV images
were post-processed in Davis 8.4 software to obtain the velocity and vorticity
fields that is subsequently utilized to construct FTLE fields and locate
vortex cores using $\Gamma_{2}$ method.

Additionally, for PLIF experiments a small amount of rhodamine 6G dye was
mixed with the fluid used to generate the vortex ring and the emitted light
was captured using the same camera mentioned above at 1000 Hz. A band-pass
filter of 570 ± 10 nm (suitable for the emission range of fluorescent dye) was
attached in front of the camera lens (LaVision) to block the scattered and
stray light from entering the camera sensor.

###  2.3 Lagrangian coherent structures (LCS) using FTLE

In the present work, the finite-time Lyapunov exponents (FTLE) analysis is
used to extract the LCS to study the interaction and mixing of the emerging
jets from the perforated surface on interaction with vortex ring. The FTLE
fields are directly calculated using the velcoity field obtained from the PIV
data. FTLE is a scalar field that is used to measure at each point in space,
the rate of separation of neighbouring particle trajectories initialized near
that point (Haller, 2001). The method gained popularity in the last decade and
is well documented (Haller, 2000; Shadden et al., 2005; Haller, 2015). Under
the action of a flow map ${F_{t_{0}}^{t_{f}}}$ the fluid particle x(${t_{0}}$)
and x(${t_{0}}$)+ ${\delta}$(x(${t_{0}}$)) are defined in space and time.
Using the local spatial gradients in the flow map i.e.,
${\nabla}$${F_{t_{0}}^{t_{f}}}$(${x_{0}}$), the Cauchy-Green tensor is
constructed (Haller & Sapsis, 2011). The coefficient of expansion or the
measure of strain is then defined as

$\sigma(x_{0},t_{0})={\lambda_{max}}([\nabla{F_{t_{0}}^{t_{f}}}({x_{0}})]^{\otimes}[\nabla{F_{t_{0}}^{t_{f}}}({x_{0}})])$
(1)

where $\otimes$ refers to the transpose of the matrix and $\lambda_{max}$
refers to the largest eigenvalue. The FTLE field is then defined as

$FTLE(x_{0},t_{0})=\frac{1}{T}\log{\sigma({x_{0},t_{0}})}$ (2)

Figure 2: (a)A 3-dimensional schematic depicting various parameters of the
vortex ring and ejection pipe (b) Dimensions of the two perforated surfaces
used in the present study

(For more details see (Shadden et al., 2005)). The maximum values of FTLE
corresponds to the LCS in the flow field. We calculate the FTLE in reverse
time known as the backward FTLE that is associated with the convergence of the
fluid particles. Higher values of FTLE means higher degree of convergence. The
integration time T can be chosen based on the extent the LCS needs to be
captured (Shadden et al., 2007) however the trajectories should not leave the
velocity field domain. We choose different values of T for different frame
cautiously keeping a check on the particle not getting accumulated at the
perforated surface and not leaving the flow domain. Further, the value of T
should not be kept smaller than the time scales of the formation of the
coherent structures. Finer spatial grids results in better and sharper fields
(Vetel et al., 2009; Espa et al., 2012). In the present work, the grid
resolution was increased to obtain sharper LCS ridges. The FTLE ridges can be
successfully utilized to identify boundaries of the coherent structure in both
laminar or turbulent flows like oceanic flows (Olascoaga et al., 2013), large
scale environmental flows (Olascoaga & Haller, 2012), mixing in turbulent
flows (Mathur et al., 2007), laminar flow structures like periodic shedding of
vortices behind bluff bodies (Kasten et al., 2010), vortex ring formation in
heart (Espa et al., 2012) and more (Haller, 2015).

###  2.4 ${\Gamma_{2}}$ method for vortex core identification

The use of vorticity fields to detect vortices can lead to erroneous
interpretations (Hussain & Jeong, 1995). Since, vorticity calculation involves
velocity gradients, the vorticity fields can fail to differentiate between
shear layers and rotating regions. In the present work, we use vorticity
contours to calculate the circulation and ${\Gamma_{2}}$ method to educe the
centres of rotating vortices. ${\Gamma_{2}}$ method, as proposed by Graftieaux
et al. (2001) is defined as

$\Gamma_{2}(P)=\frac{1}{N}\sum_{S}\frac{({\vec{PQ}}\times({\vec{U_{Q}}-\vec{U_{m}}}))\cdot\vec{z}}{\parallel\vec{PQ}\parallel\cdot\parallel\vec{U_{Q}}-\vec{U_{m}}\parallel}$
(3)

where the formulation is applied at each point P in the vector field obtained
from the PIV data to find the normalized scalar $\Gamma_{2}$ values.
$\vec{PQ}$ denotes the displacement vector from point P to Q inside area S,
$\vec{U_{m}}$ refers to the mean velocity of the area S, $\vec{U_{Q}}$ refers
to the velocity vector at point Q and $\vec{z}$ is the unit vector
perpendicular to the plane. The threshold for the vortex centre is considered
to be 0.75 (i.e., $>$ 2/$\pi$) since the structures form in the downstream
region after interaction are turbulent and chaotic in nature. The
consideration of the local velocity vector $\vec{U_{m}}$ makes the method
Galilean invariant unlike $\Gamma_{1}$ method (Graftieaux et al., 2001).

Pin | rc | R | Uc | ${\Gamma_{0}}$ | ReΓ | Reconv
---|---|---|---|---|---|---
(psi) | (cm) | (cm) | (cm/s) | ($cm^{2}/s$) | (${\Gamma_{0}/\nu}$) | (${U_{c}R/\nu}$)
40 | 0.36 $\pm$ 1% | 0.8 $\pm$ 1.6% | 30 $\pm$ 3% | 115 $\pm$ 1.1% | 11500 | 2400
Table 1: Different parameters of the vortex ring. The $\pm$ % indicates the
positive and negative standard deviation values of the corresponding mean
values. Here, Pin refers to the injection pressure at which the vortex ring is
generated inside the water; rc refers to the radius of the core of the vortex
ring (check figure 2(a)); R is the distance between the vortex centre to the
core known as the radius of the vortex ring (check figure 2(a)); Uc is the
convection velocity of the vortex core in y direction calculated by tracking
the vortex core; ${\Gamma_{0}}$ is the circulation of the free vortex ring
i.e., the case without any perforated plate; ReΓ is Reynolds number based on
circulation; Reconv is the Reynolds number based on convection velocity and
ring radius.

###  2.5 Experimental conditions and measurements

The experiments are conducted under water at room temperature. For all the
values of ${\phi}$ and ${\theta}$ the distance between the ejection pipe and
the apex of the perforated surface was kept at 7D. This distance is kept such
that the vortex ring is fully developed during the start of the interaction.
It is worth noting that in the present scenario, at lower ${\theta}$, the
vortex ring will start interacting earlier compared to larger ${\theta}$
values and hence the apex is considered as the reference point from which the
y axis passes. This axis also coincides with the translational axis of the
vortex ring. The various specifics of the vortex ring are enumerated in Table
1 and has been depicted in Figure 2(a). The velocity and vorticity
(${\omega}$) fields obtained from the PIV measurements are used to calculate
the circulation (${\Gamma}$) by taking the area integral of the vorticity
field. The formulation for circulation is given as

$\Gamma=\int_{C}\omega dA$ (4)

where C is the area of interest. For all the cases, a threshold of 10% of the
maximum vorticity was considered to calculate the circulation which was found
to be sufficient to eliminate the background noise in the PIV data. For
calculating the circulation, a masking region was defined as shown in Figure
1(c) encompassing the desired events.

##  3 Results and Discussion

###  3.1 Characterization of the vortex ring

Vortex ring having ReΓ = 11500 (at 5D from ejection pipe) is considered in the
present study to understand how ${\theta}$ and ${\phi}$ influences the
interaction. Figure 3(a) depicts the variation of ${\Gamma^{+}_{0}}$ and
${\Gamma^{-}_{0}}$ (circulation generated by positive and negative vorticity
respectively) of the free vortex ring (without perforated surface) with  T∗
(non-dimensionalised using the ratio ${U_{c}/D}$). This period corresponds to
a physical distance of $\sim$ 3 cm (2.5D) starting at 5D from the ejection
site. The constant value of circulation confirms the fully developed condition
of the vortex ring before the interaction. Figure 3(b) shows the y velocity
profile of the vortex ring along the line joining the two cores as shown in
the inset. The profile obtained corresponds to a typical velocity profile
across the vortex ring (Maxworthy, 1977). From the profile, the symmetric
nature of the vortex ring can be confirmed. Figure 3(c) depicts the velocity
vector field of a propelling vortex ring superimposed over the vorticity
contours. The centres of the two cores are identified using the $\Gamma_{2}$
method as described in section 2.4. It is clear that the peak vorticity and
the centre do not coincide. The backward time FTLE field of the vortex ring is
laid over the vector field in figure 3(d). The bright ridges corresponding to
the higher FTLE values demarcating the vortex ring boundary across which
material transport remains restricted.

Figure 3: (a) Variation in positive and negative circulation of the free
vortex with the non-dimensional time (${T^{*}=\textit{${U_{c}/D}$}}$). The
distance covered during this time period is starts from 5D till 7.5D from the
ejection pipe (b) The y-velocity ($V_{y}$) profile of a free vortex at 5D
distance form the ejection pipe along the line passing through the vortex core
as shown in the inset (c) Vorticity contour plot of a free vortex ring
overlaid over velocity vectors field. The pink and green dot represents the
counter-clockwise (CCW) and clockwise (CW) vortex core detected using
$\Gamma_{2}$ method (d) The backward FTLE field of a free vortex overlaid over
the velocity vector field.

###  3.2 Qualitative aspects of the interaction

To develop an initial understanding of the interaction, we show in figure 4
and 5 the time sequenced PLIF images at different non-dimensional times (T∗=
${tU_{c}/D}$)) for all the ${\theta}$ values at ${\phi_{1}=0.24}$ and
${\phi_{2}=0.44}$. T∗= 0 is consider just before the dyed vortex collides with
the perforated wall. Due to hindrance from the perforated surface, the left
side of the interaction process is not visible or partly visible hence, the
right side is considered for discussion. An important aspect of increasing the
${\theta}$ values is that it delays the interaction phenomenon and shifts the
interaction site further downward. This combined with the ${\phi}$ has a
direct consequence on the number of holes involved during the interaction
process. For ${\phi_{1}\theta^{60}}$ (Figure 4), the vortex ring on
interaction starts emerging out as unstable jets from the $4^{th}$ hole
counted from the apex. This number decreases by 1 till
${\phi_{1}\theta^{120}}$ after which it emerges from 2 holes. For $\phi_{2}$,
the number of holes remain 5 for ${\phi_{2}\theta^{60}}$ that decreases by 1
till ${\phi_{1}\theta^{150}}$. This is because of smaller pitch between holes
for ${\phi_{2}}$ compared to ${\phi_{1}}$ (check figure 2). These observations
will be quantitatively explained in further sections. For both
${\phi_{1}\theta^{180}}$ and ${\phi_{2}\theta^{180}}$ the vortex ring comes
out of 5 holes however, shifting the position of the perforated surface for
${\phi_{2}\theta^{180}}$ results in involvement of 6 holes. For consistency,
we ignore this case in the present study and consider the case when the
translational axis of the vortex ring coincides with one of the hole centres
for ${\phi_{2}\theta^{180}}$.

Figure 4: The PLIF images for ${\phi_{1}}$ at ${\theta}$ = 60° - 180° shown at
different ${T^{*}}$. The scale represents 5 mm.

Figure 5: The PLIF images for ${\phi_{2}}$ at ${\theta}$ = 60° - 180° shown at
different ${T^{*}}$. The scale represents 5 mm.

Figure 6: The reformation of the vortex ring in the far downstream region
($\sim$ 6D from the apex). The red box in case of $\theta$=150° depicts the
suppression of the reformed vortex ring. The scale bars represents 5 mm.

On interacting, the vortex ring breaks down into circular jets that develop
instability over very small time scales. Due to the quiescent nature of fluid
downstream of the surface, Kelvin-Helmholtz (K-H) instabilities grows at the
shear layers of the jets that are clearly visible in the figures (4-5). For
cases with lowest $\theta$ value (i.e., 60∘), the flow after interaction has a
high tendency to curl in the same sense as the right hand part of the vortex
ring (as will be discussed in section 3.3.2, check figure 14). The emerging
jets starts travelling in x direction with their heads rotating in an opposite
sense. This can be clearly seen in ${\phi_{2}\theta^{60}}$ (Figure 5) where
the head of the jets roll up to form series of circular pattern, a feature
very commonly observed during the occurrence of K-H instabilities.
Furthermore, for ${\phi_{1}}$ the out-coming jets maintain a distance with the
adjacent jets for a long time before interacting as is visible in Figure 4
(${\phi_{1}\theta^{60}}$ : T∗=1.1, ${\phi_{1}\theta^{90}}$:T∗=1.3, 1.6,
${\phi_{1}\theta^{120}}$:T∗ = 0.8, ${\phi_{1}\theta^{150}}$: T∗=1.1, 1.7,
${\phi_{1}\theta^{180}}$: T∗=1). Whereas in cases with ${\phi_{2}}$, these
jets interacts with their adjacent jets very early because of the proximity of
the holes. These early interactions makes the downstream flow highly chaotic
in nature. Adding on to the instabilities, we observe an intriguing phenomenon
that results in the mushroom shaped features at the head of the jets that will
be discussed in section 3.2.1.

The jets after ejecting out self interacts resulting in a violent mixing of
the fluid that has been observed in previous studies (Xu et al., 2018; Jain et
al., 2023). This interaction is characterized by 3-dimensional mixing and
vorticity cancellation. The consequence of vorticity cancellation is the
formation of two primary coherent structures in the downstream with low
strength . With increasing ${\theta}$, the fluid from the two halves of the
surface starts to approach each other and interacts for ${\theta}$=150°
(figures 4 and 5, $\theta$=150°). The distance at which the two halves
interact is not sufficient for each half to form the downstream vortex hence,
we did not observe collision of two downstream regenerated vortex. Due to this
type of interaction, no coherent structure is observed for
${\phi_{1}\theta^{150}}$ cases. For all the other cases the regenerated
downstream vortex is shown in Figure 6.

It can be understood that on interaction, the vortex ring expands in the z
direction as well as has been discussed by New et al. (2020) in details. In
their study, the walls had no penetration condition which forced the vortex
ring to expand in the z direction. In the present study due to the change in
the penetration condition at the wall, we see a rapid passage of vortex core
through the holes in a sequential manner as discussed above. An image (see
supplementary figure S1) has been provided to illustrate the amount of dyed
fluid ejecting of the holes present in the z direction from the apex for
${\phi_{1}\theta^{90}}$. The image shows that for lower open area ratio i.e.,
${\phi_{1}}$, the amount of fluid coming out at different distances remains
significantly less. Essentially, this must depend strongly on the ${\phi}$
value of the system and is expected to be much more lesser for ${\phi_{2}}$.

####  3.2.1 Initial instability on jets

It is well known that vortex ring induces fluid motion around themselves. The
velocity field around a vortex decreases gradually as shown in figure S2 and
the induced velocity field around the perforated surface for $\theta$ = 60° is
shown in figure S3. Due to the induced flow and the presence of perforated
surface of appreciable thickness, the fluid ahead of the vortex ring crosses
the perforated surface much before the vortex ring does. As the induced flow
passes through the holes, the shear resistance offered by the holes leads to
the development of mushroom shape front. This can be seen in figures 7 and S4
where we capture this phenomenon for ${\theta}$ = 150° and 180° respectively
by pouring a thin layer of rhodamine dye over the perforated surface just
before the vortex ring approaches it. This front is further caught up by the
main flow from the vortex which can be seen from the marked images (figures 7
and S4). The initial jets have low kinetic energy as the peak comes later (T∗
$>$ 1, check figure 8) due to penetration of the core and hence this initial
jet converges to the shape of this mushroom. This imparts the initial
instability to the jet heads and they form a mushroom shaped feature.In case
of ${\phi_{2}}$, these mushrooms remains at close proximity tending to overlap
at the edges thereby forming a continuous structure (figure 7(b) and S4),
unlike in ${\phi_{1}}$ where the boundaries of the mushrooms are distinctly
detectable. As the core starts to penetrate, velocity peak is obtained and
intense mixing between jets is observed along with K-H instability at the
edges of the jets. The initial instability discussed here adds on to the
rolling up of the jets at their heads. This type of mushroom formation is a
direct consequence of placing a perforated plate with considerable thickness
in the path of a vortex ring.

###  3.3 Quantitative aspects of the interaction

####  3.3.1 Evolution of velocity and vorticity field near the surface

As seen in the preceding sections, the vortex ring on colliding with the
inclined perforated surface emerges out in a sequential manner. To better
understand the phenomenon we plot the non-dimensional velocity magnitude
(${V_{mag}^{*}}$, normalised using the velocity magnitude of the free vortex)
and the dimensional vorticity values about a line at a distance of $\sim$ 0.5D
from the perforated surface in the downstream region as shown in figure 8 and
9 respectively. This distance was considered to avoid the contamination of the
data by the noisy signals generated near the wall. Also, beyond this distance,
some of the peaks were lost for few cases. In the y axis the non-dimensional
length ($L^{*}=L/D$) with 0 being the apex point is plotted along with the
non-dimensional time ($T^{*}=tU_{c}/D$) in the x axis. $T^{*}=0$ is considered
the frame just before the velocity vectors starts emerging from the perforated
surface in the PIV data. The primary purpose of these plots is to depict the
sequential generation of jets form the perforated plate and to capture the
near wall events when the jets emerge out. For ${\phi_{1}\theta^{60}}$, we see
a weak $V_{mag}^{*}$ peak followed by three dominant peaks occurring one after
the other as time progresses (figure 8). Although for ${\phi_{1}\theta^{60}}$
the vortex interacts early with the wall, the initial impact remains weak and
vortex rapidly gets sucked in towards the apex. The initial interaction does
not involve the disintegration of the vortex core resulting in a feeble peak.
Overall, we see three major peaks for this case that slowly convects out with
time. Similarly for ${\phi_{2}\theta^{60}}$, the interaction starts even
earlier which is expected and 5 distributed peaks are obtained in a successive
manner (figure 8). The phenomenon seen in the PLIF images (figure 4 and 5) for
${\theta}$=60° has been beautifully captured by the PIV data as can be seen
form the vorticity values plotted in figure 9. For ${\phi_{1}\theta^{60}}$ we
can see the appearance of alternate sense of vorticity patches i.e., positive
vorticity (red contours, CCW) sandwiched between negative vorticity zones
starting from the farthest hole with weak negative vorticity zone. Whereas for
${\phi_{2}\theta^{60}}$, we can see a continuous stretch of CW vortices
appearing initially followed by slightly shifted stretch of CCW vortices. The
vorticity zone near the apex maintains its vorticity for the longest period
due to the decay of the vortex near apex.

Figure 7: The PLIF images for ${\phi_{1}\theta^{150}}$ and
${\phi_{2}\theta^{150}}$ depicting the initial jet instability due to mushroom
formation. Here, $\Delta T^{*}$ is the difference in $T^{*}$ The scale
represents 5 mm.

Figure 8: The non-dimensional velocity magnitude (non-dimensionalised using
free vortex ring velocity magnitude) contour plotted against ${T^{*}}$ and
${L^{*}}$ (= $L/D$ where L is the length of the line about which the data is
plotted) at a distance of 0.5D from the perforated surface.

Figure 9: The dimensional vorticity contour plotted against ${T^{*}}$ and
${L^{*}}$ (= $L/D$ where L is the length of the line about which the data is
plotted) at a distance of 0.5D from the perforated surface.

For ${\theta}$ = 90°, three velocity peaks are seen followed by two peaks for
the rest of the cases (figure 8). For cases with ${\phi_{1}}$, a distinct gap
between the peaks can be observed that is not present in case of ${\phi_{2}}$.
For ${\phi_{2}\theta^{150}}$, the peaks are seen to be merging with each
other. Another important information that can be extracted from the plot is
that for ${\theta}$ $\geq$ 120° , the first velocity peak occurs nearer to the
apex and the second peak occurs away from it i.e., there is reversal in the
sequence in which the jets come out (figure 8). This suggest that the core
first starts to penetrate from the holes nearer to the apex of the perforated
surface. Hence, for the present set of parameters, ${\theta}$ = 120° behaves
like a transition point where the flow field changes and starts to tend
towards $\theta$ = 180°. Moreover with increase in ${\theta}$, the overall
flow map (figure 8 and 9) can be seen to be shifting towards $L^{*}$ = 0. For
${\phi_{2}}$, the vorticity zones are more circular in shape compared to
${\phi_{1}}$ where they are more linear. The overall vorticity dynamics will
be discussed in the next section.

####  3.3.2 Vorticity dynamics

Figure 10: Evolution of vorticity field for ${\phi_{1}\theta^{60}}$ and
${\phi_{2}\theta^{60}}$ plotted at different $T^{*}$ values. The scale vector
indicates $1m/s$. The pink dot represents the CCW rotating vortex cores and
the green dots represents the CW rotating vortex cores detected using
$\Gamma_{2}$ method.

Figure 11: Evolution of vorticity field for ${\phi_{1}\theta^{90}}$ and
${\phi_{2}\theta^{90}}$ plotted at different $T^{*}$ values. The scale vector
indicates $1m/s$. The pink dot represents the CCW rotating vortex cores and
the green dots represents the CW rotating vortex cores detected using
$\Gamma_{2}$ method.

We plot the vorticity contours over the vector fields obtained from the PIV
data in figures 10-14 for different ${\theta}$ values in the increasing order.
When a vortex ring interacts with a perforated plate of such kind, the
emerging flow in the downstream remains highly chaotic. The reason for this
being the conversion of the vortex core to single jets and their further
interaction with each other. For cases with ${\theta}$ $\leq$ 120° , we see
this phenomenon occurring symmetrically on two halves of the surface whereas
for ${\theta}$ $>$ 120 the jets on both sides starts talking to each other.
These features will be highlighted in this section where we discuss how the
vortex cores move and interact in the downstream region. We denote the
clockwise (CW) and counter (CCW) rotation of vortices by their centre using
green and pink dots respectively. The location of the dots have been
identified using the $\Gamma_{2}$ method as described in section 2.4.

Figure 12: Evolution of vorticity field for ${\phi_{1}\theta^{120}}$ and
${\phi_{2}\theta^{120}}$ plotted at different $T^{*}$ values. The scale vector
indicates $1m/s$. The pink dot represents the CCW rotating vortex cores and
the green dots represents the CW rotating vortex cores detected using
$\Gamma_{2}$ method.

For ${\phi_{2}\theta^{60}}$ we see a stretch of negative vorticity coming out
of the surface compared to ${\phi_{1}\theta^{60}}$ where the features
resembles more like individual jets (figure 10). A CCW vortex develops near
the hole that is farthest to the apex. The vorticity remains more distributed
in case of ${\phi_{2}\theta^{60}}$ and the flow moves at a slower speed. For
${\phi_{1}\theta^{60}}$ ultimately the flow converges to the development of
two major vortical regions that propel in the direction nearly perpendicular
to the perforated surface. For ${\phi_{2}\theta^{60}}$, the CW vortex that
develops near the apex has more vorticity and propels faster than the CCW
resulting in the horizontal convection of the flow. The faster motion of this
CW vortex is a result of the final momentum that is is imparted by the
disintegration of the core near the apex. The two vortical regions at the
extreme ends of the flow is seen to break up into multiple small scale
vortical zones. Similar is the case for ${\theta}$ = 90°(figure 11) where the
rotating zones develop very early and propels in a nearly normal direction.

Figure 13: Evolution of vorticity field for ${\phi_{1}\theta^{150}}$ and
${\phi_{2}\theta^{150}}$ plotted at different $T^{*}$ values. The scale vector
indicates $1m/s$. The pink dot represents the CCW rotating vortex cores and
the green dots represents the CW rotating vortex cores detected using
$\Gamma_{2}$ method.

We depict both the sides of the flow for ${\theta}$ $\geq$ 120°to identify the
$\theta$ at which the flow on the two halves starts to interact with each
other. Jets of opposite vorticity can be seen to develop initially around
$T^{*}$ = 0.65 for ${\phi_{1}\theta^{120}}$ with the rotating regions on the
outer and inner sides of the flow on each side (figure 12). For
${\phi_{2}\theta^{120}}$ instead of long jets, we see circular patches of
vorticity developing initially around the same time. The out coming flow in
${\phi_{2}}$ already starts to interact due to the proximity of the holes
resulting in larger circular patches. Interestingly, the vorticity
cancellation process seems less intense in these larger vorticity zones (in
${\phi_{2}}$) compared to the jets (in ${\phi_{1}}$) resulting in the higher
vorticity values of the core at a later instance. Due to a larger spreading
area in case of ${\phi_{2}\theta^{120}}$, we can see at ${T^{*}}$ = 3.4, the
inner vortical cores come very near to each other ultimately diverging in the
flow direction.

Figure 14: Evolution of vorticity field for ${\phi_{1}\theta^{180}}$ and
${\phi_{2}\theta^{180}}$ plotted at different $T^{*}$ values. The scale vector
indicates $1m/s$. The pink dot represents the CCW rotating vortex cores and
the green dots represents the CW rotating vortex cores detected using
$\Gamma_{2}$ method.

For cases with ${\theta}$ = 150°(figure 13), the flow initially behaves
similarly to the preceding case with much more proximity between the inner
jets from other halves. Along with vorticity cancellation on each sides of the
surface, we see clear interaction of the vortex cores for
${\phi_{1}\theta^{150}}$ at ${T^{*}}$ = 3 and ${\phi_{2}\theta^{150}}$ at
${T^{*}}$ = 2. Neat jets of fluid can be discerned for
${\phi_{1}\theta^{150}}$ that results in vorticity cancellation from adjacent
jets and ultimately breaks into multiple vortical zones as shown in figure 13.
However, a less coherent flow is observed for ${\phi_{2}\theta^{150}}$
resulting in the development of many vortical zones resembling turbulent
flows. As shown in figure 6 (red marked zone), the vortex reformation in the
far downstream does not happen in case of ${\theta}$ = 150° due to the
interaction of flow from other halves.

For ${\theta}$ = 180°(figure 14) we show both the upstream and the downstream
regions of the perforated surface. For other cases the upstream flow was not
depicted because of the uncertainty in the PIV data that can arise in cases
with lower ${\theta}$ values due to wall effects. In this regard, a discussion
and plot (figure S5) is presented in the supplementary sheet that shows the
effect of $\phi$ and $\theta$ on the circulation of the vortex ring just
before the interaction. On the upstream side, as the vortex approaches the
surface boundary layer of opposite sense starts to develop that separates to
form the secondary vortices of opposite sense (Naaktgeboren et al., 2012; Jain
et al., 2023). It can also be seen that the vortex core sustains for a longer
period in case of $\phi_{1}$ which is expected due to lesser porosity. For
${\phi_{1}\theta^{180}}$, the jets of opposite vorticity interact and shed
vortices from the head in the central region as can be seen in figure 14. The
central vortical structure is a result of the vorticity cancellation and the
unstable head of the jets due to the initial mushroom type instability as
discussed in section 3.2.1. These central vortices are also seen for
${\phi_{1}\theta^{150}}$ at ${T^{*}}$ = 3 due to the interaction of the
central jets and their shedding. Along with theses small structures, the
overall flow preserves the axial bulk velocity at the central region resulting
in the formation of two large vortical cores at the edges of the flow field
(Jain et al., 2023). In contrast, we do not observe central vortices for
${\phi_{2}\theta^{180}}$ and two large scale vortical structures form after
the central vortices cancel each other.

A quantitative comparison of the non-dimensional circulation generated by the
positive and negative vorticity in the downstream region is presented in
figure 15 for each of the cases. Owing to symmetry, we consider the right side
of the apex axis for all the cases (as shown in figure 1(c)). The pink line
depicts the circulation generated by the positive vorticity ($\Gamma^{+}$,
CCW) and the green line shows the circulation generated by the negative
vorticity ($\Gamma^{-}$ ,CW). Since the right hand side of the flow is
considered, the side of the vortex ring having CCW sense about the XY plane is
discussed. The fluid emerging out innately has a natural tendency to generate
positive circulation in CCW direction. This is observed in all the cases with
the pink line always having higher value compared to the green line. In figure
15, for $\theta$=180° as well the right hand side of the flow is considered.
Figure S6 has been provided having the positive and negative circulation for
the whole field depicting the symmetricity of the interaction process. For
$\theta$ = 60° initially, the flow is highly dominated by the positive
vorticity in (especially ${\phi_{2}\theta^{60}}$) which further subsides to
constant values. The trend in case of $\theta$ = 90° and $\theta$ = 120°
almost remains similar. It can be inferred from the graph that the ratio of
positive circulation to the absolute of negative circulation for
$\theta$$\leq$ 120° becomes constant (near $\sim$ 1) after a point of time
indicating the development of two coherent vortex structures of similar
circulation without any further generation of circulation. Interestingly, for
$\theta$ = 120° the pink and green line remains very near and behaves similar
to what is shown in figure S6 for the full flow field of $\theta$ = 180° .
Further, the difference between the peaks goes on decreasing till $\theta$ =
150° where we see the peaks of around similar strength coming out of the one
side of the surface. However, intriguingly we observe that the green and pink
line does not converge any more as seen in cases below $\theta$ $\leq$ 120° .

Figure 15: The variation of absolute values of non-dimensional positive and
negative circulation (non-dimensionalised by $\Gamma_{0}$ for each case) in
the downstream region of the flow field after interaction for all the cases
with $T^{*}$. Here, $T^{*}$ = 0 represents the instance at which the velocity
vectors becomes detectable after interaction in the downstream region. SD
represents standard deviation of the data.

Figure 16: The variation of the ratio of absolute values of positive and
negative circulation in the downstream region of the flow field after
interaction for the mentioned cases with ${T^{*}}$. Here, ${T^{*}}$ = 0
represents the instance at which the velocity vectors becomes detectable after
interaction in the downstream region.

To further investigate this divergence or non-convergence we plot circulation
ratio ($\Gamma^{+}/\Gamma^{-}$) against the non-dimensional time ($T^{*}$) in
figure 16. We see that for both ${\phi_{1}\theta^{150}}$ and
${\phi_{2}\theta^{150}}$ the ratio of circulation starts to increases from the
point where it nears the value of 1. This increase is more dominant and
happens earlier for ${\phi_{2}\theta^{150}}$. The increase in the ratio of
circulation after achieving the peak indicates the generation or an imbalance
in vorticity because of the interaction of flows from the either sides of the
permeable surface. The interaction as observed in figure 13 results in the
generation of multiple smaller vortical structure that contributes to enhanced
vorticity and subsequent circulation. The loss of coherency in the downstream
flow because of the proximity of the holes in case of $\phi_{2}$ cause the
circulation ratio to increase before reaching the value 1. For cases with
$\theta$ = 180°, the occurrence of negative vorticity (figure 15) is because
of the opposite vorticity of one side of the jet occurring on the right side
of apex that attains a much weaker peak before disintegrating due to vorticity
cancellation. The value of negative circulation is higher in case of
${\phi_{1}\theta^{180}}$ compared to ${\phi_{2}\theta^{180}}$ at later stages
because of the generation of vortices at central region (as discussed in
section 3.3.2). However, a larger value of positive value is a result of
larger vortex structure seen in case of ${\phi_{2}\theta^{180}}$.

Figure 17: A general schematic diagram depicting the chain of events and the
generation of reformed vortex by the interaction of single half of the vortex
ring. The figure is not to scale is meant for qualitative description.

Figure 18: Variation of the non-dimensional negative circulation peak in the
downstream region with $\theta$.

We see that a single half of vortex ring on interaction with a single half of
the perforated surface results in the formation of a full vortex ring (figure
6). A schematic to understand this phenomenon has been provided in figure 17.
As we consider the right hand side of the interaction zone, the CCW part of
the vortex ring interacts with the perforated surface to generate vorticity of
both CW and CCW to ultimately reform a vortex ring. Although, the values of
positive circulation (CCW) remains higher (check figure 15), the value of
negative circulation is strongly dependent on the $\theta$ value. Figure 18 is
plotted to show the variation of negative circulation peak for both the values
of $\phi$ at different values of $\theta$. It can be seen that for $\phi_{2}$,
the strength of the negative circulation generated is consistently lesser for
all values of $\theta$ compared to that in $\phi_{1}$. With increase in the
value of $\theta$, the normalized negative circulation increases that reaches
a peak at $\theta$ = 150° before falling significantly at $\theta$ = 180°
(considered a half part). The fall is expected since for $\theta$ = 180° we
see a very symmetric nature of the circulation produced when the whole
downstream flow field is considered (figure S6). The negative circulation
region shifts towards the central region which is not fully considered here.
Further analysis of the reformed vortex ring can provide deeper insights into
the symmetricity and contribution of the negative circulation on its overall
strength.

To summarize the vorticity dynamics, we observe a more distributed flow field
in case of ${\phi_{2}}$ compared to cases with ${\phi_{1}}$. The jet like
features coming out of ${\phi_{1}}$ and their interaction results in the
shedding of multiple small vortical features. The vorticity cancellation seems
to be less intense in case of ${\phi_{2}}$ because of the generation of
dominant vorticity patches at the outer edges of the flow and smaller patches
at the central region. However, it is difficult to precisely quantify the
vorticity cancellation phenomenon owing to the 3-dimensional nature of the
flow downstream. For all the cases except for $\theta$=150° where the fluids
from the two halves of the surface interacts with each other, we observe the
formation of two primary vortical structures that propel in the direction
roughly normal to the surface (except for ${\phi_{2}\theta^{60}}$). The
interaction between the flows between the two halves results in the creation
or imbalance of vorticity from the flow shearing between different rotating
structures. What we also infer is that by keeping a perforated surface with
60°$\leq$$\theta$$<$150° of the kind used in the present work ($d_{h}$ = h), a
vortex ring is split in to two vortex rings that propels in the direction
imposed by the plate inclination. The process involves the generation of jets
of opposite vorticity that results in the reformed vortex. Such arrangements
can be used to manipulate flow at various scales. Further, a value near to
$\theta$=150° can be utilized to suppress the re-formation of the vortex ring
after interacting with perforated plates.

Figure 19: Backward FTLE field for (a) ${\phi_{1}\theta^{60}}$ and (b)
${\phi_{2}\theta^{60}}$ shown at different $T^{*}$ corresponding to figure 10.

####  3.3.3 Lagrangian analysis

Figure 19, 20, 21 and S7 shows the backward FTLE fields for $\theta$ = 60° ,
120° 150° and 180° respectively. FTLE fields are shown at the same $T^{*}$ for
which the vorticity contours are plotted (in figures 10, 12, 13 and 14). Our
primary aim is to elucidate if the jets emerging out of the permeable surface
mix at all and if it does, how and when. Although, a lot has been discussed in
details about the interaction of jets in the above sections, we finally want
to conclude our analysis with a qualitative evaluation of the FTLE fields
digging deeper into the dynamics of out coming jets from the holes. The LCS
are the ridges of the FTLE fields that is transverse to the direction of
minimum curvature (Haller, 2001). Based on the system’s dynamical behavior,
the LCS in the flow fields represent surfaces that does not allow transport
across it. We plot the FTLE values by integrating backward in time that yields
attractive FTLE ridges. Since, we use different integration time for each of
the $T^{*}$, the FTLE range obtained for each of the frame varies. However,
bringing out the bright ridges to understand the mixing dynamics remains the
central theme of this exercise.

Figure 20: Backward FTLE field for (a) ${\phi_{1}\theta^{120}}$ and (b)
${\phi_{2}\theta^{120}}$ shown at different $T^{*}$ corresponding to figure
12.

Figure 21: Backward FTLE field for (a) ${\phi_{1}\theta^{150}}$ and (b)
${\phi_{2}\theta^{150}}$ shown at different $T^{*}$ corresponding to figure
13.

From the first instance, brighter ridges between the jets coming out of the
perforated surface can be detected in case of ${\phi_{1}\theta^{60}}$ unlike
that for ${\phi_{2}\theta^{60}}$ (figure 19) where a continuous front of the
attractive LCS develops. The ridge front of the flow structure is brightest at
the farthest side of the apex due to the early emergence of flow for
${\theta}$ = 60° as discussed earlier. For ${\phi_{1}\theta^{60}}$, the
separation between the jets can be clearly observed at $T^{*}$ = 1.5 that
further starts to blur out with the passage of time. However, for
${\phi_{2}\theta^{60}}$ we see very less amount of bright separatrices
appearing between the jets indicating towards the mixing of flow at very early
stages of emerging out. The flow structures at the later stages corresponds
well with the vorticity contours in figure 10. Similar patterns can be
observed for ${\theta}$ = 120°(figure 20) with highly mixed flows for
${\phi_{2}\theta^{120}}$. We see no interaction between the ridges of the
other halves of the surface and neat vortical structures develop. In case of
${\phi_{1}\theta^{150}}$ (figure 21), the interaction of the flow from the
other halves can be seen around $T^{*}$ = 3 whereas this begins much early for
${\phi_{2}\theta^{150}}$. At further instances, very chaotic flow with random
distribution of attractive ridges develops which was educed from PLIF and PIV
results. Finally, with the attractive LCS we could successfully explain the
various events taking place during the interaction process. Although we do not
show the forward FTLE or the repelling ridges, it can effectively be utilized
to understand the entrainment quality associated with the flow structures that
reform in the downstream region after interacting with the perforated surface.
Furthermore, a rigorous quantitative analysis of the ridges can bring out more
insights into the mixing phenomenon among the jets.

##  4 Summary and Conclusions

Experiments were conducted to explore the interaction dynamics of a vortex
ring ($Re_{\Gamma}$ = 11500) impinging on perforated surface (${\phi_{1}}$ =
0.24 and ${\phi_{2}}$ = 0.44) with different included angles (${\theta}$ =
60°- 180°) using PLIF, PIV, Lagrangian FTLE and $\Gamma_{2}$ vortex
identification techniques. By changing the the value of $\theta$ the
interaction process is altered significantly. The early interaction of the
vortex ring with the perforated surface results in a sequential ejection of
the fluid and a reduction in number of ejecting holes with increase in the
${\theta}$ value, a phenomenon very well captured by the PLIF images. Due to
higher proximity of the holes in case of ${\phi_{2}}$, the jets are seen to
interact immediately as it start to emerge out of the perforated surface.
Whereas for ${\phi_{1}}$, the jets interact much later and their development
is characterized by the occurrence of K-H instability at its edges.

An interesting phenomenon involving the development of mushroom shaped
features is reported. The fluid ahead of the vortex ring moving with the
induced velocity crosses the perforated surface earlier than the main vortex
resulting in the growth of dormant mushroom structures which is later caught
by the faster main flow. These structures are a result of placing a perforated
plate in the path of a vortex ring and imparts initial instability to the jet
structures.

From the time series contour plot of $V_{mag}^{*}$, we show that the
sequential ejection of the vortex ring in the forms of jets. Further, a change
in the slope in the $V_{mag}^{*}$ peak for $\theta$ $>$ 90° with time indicate
towards the shift of the first ejection hole of the vortex ring. This marks a
transition point from where the flow starts to tend towards ${\theta}$ = 180°.
The vorticity dynamics of the flow for all the cases has been discussed in
details supported by the vortex centres that are identified using the
$\Gamma_{2}$ method. The formation, interaction and cancellation of the
vortical structures both in the form of jets and circular patches are shown.
The movement of the cores and their interaction at $\theta$ = 150° is shown
along with the generation of multiple small scale vortical zones. The
vorticity cancellation phenomenon occurs for both the ${\phi}$ values but
differently, which ultimately results in the preservation of the velocity
profile of the vortex ring leading to the reformation of vortex ring at far
downstream except in $\theta$ = 150°. Further analysis of the development of
circulation over time revealed the occurrence of an imbalance in the vorticity
due to the shearing of the flows from two halves of the surface in case of
$\theta$ = 150°. The peak of both the positive and negative circulation
achieves similar non-dimensional circulation but diverges after a point of
time and behaves like $\theta$ = 180°case suggesting the fading effect of the
effect of $\theta$. The Lagrangian FTLE fields finally confirmed towards the
sequential emerging, early interaction of flow from holes in case of
$\phi_{2}$ and and interaction of flow from other halves resulting in
turbulent structure with random attractive LCS.

Inline with the objectives identified in section 1, we have successfully
answered how the vortex ring emerges out of inclined porous surfaces and the
effect of different $\theta$ on the downstream flow. The behaviour and
interaction of jet was also studied however, a more rigorous analysis of the
emerging jets can reveal more insights into the phenomenon. Finally, the
effect of $\phi$ was seen to be significant in changing the flow dynamics.

It is worth mentioning that by placing a perforated plate with different
$\theta$ values, a reformed vortex can be generated from each side i.e., a
single vortex ring is capable of generating two reformed vortex rings.
Further, this generation of reformed vortex ring can also be suppressed by
maintaining the $\theta$ value near to 150° between the surfaces. Such results
can prove to be useful for flow control and manipulation purposes.

##  5 Acknowledgement

S. Jain and S.R Rao would like to thank the Prime Minister Research Fellowship
(PMRF) for the financial support.

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##  6 Supplementary Data

Figure 22: Figure S1: PLIF images after interaction of vortex ring with
perforated surface for ${\phi_{2}\theta^{150}}$ captured at different planes
measured from the hole aligned with the vortex ring’s translational axis. For
1.35D and beyond, the images are captured at later times since the fluid comes
out of the farther holes much later after the interaction at 0D takes place.

Figure 23: Figure S2: The velocity magnitude $V_{m}ag$ along the lines AB, AC
and AD where A corresponds to the center of the vortex ring. The angle between
AB-AC and AC-AD is 45°. The shaded region represents the standard deviation
(SD).

Figure 24: Figure S3: The induced velocity field across the perforated surface
for $\theta$=60°is shown. The left column represents the raw PIV images, and
the corresponding processed image is shown in the right column.

Figure 25: Figure S4: The PLIF images for ${\phi_{1}\theta^{180}}$ and
${\phi_{2}\theta^{180}}$ depicting the initial jet instability due to mushroom
formation. Here, $\Delta^{*}$= is the difference in $T^{*}$. The scale
represents 5 mm.

Figure 26: Figure S5: The circulation of the vortex ring just before the
interaction as shown in the inset normalized with the circulation of free
vortex $(\Gamma_{0})$. The error bars represent the standard deviation (SD).
For all the case, a minimal change in the circulation value is observed with
respect to $(\Gamma_{0})$ with a maximum reduction of around 10%. The values
are particularly lower for the case of $\phi_{1}$ at all the $\theta$ which
has a lower open area and behaves more like impermeable wall compared to
$\phi_{2}$. It is expected that larger open area cases will behave more like
free vortex case (i.e., without any surface) and the values should be nearer
to 1. As the $\theta$ value reaches 180 °, the ratio approaches the value 1
for both the case. The presence of the wall near the vortex ring before the
interaction as shown in the inset contaminates the PIV data and hence, we see
higher values of SD values.

Figure 27: Figure S6: The variation of absolute values of non-dimensional
positive and negative circulation in the downstream region of the flow field
after interaction for $\theta$ = 180°cases with $T^{*}$. Here, $T^{*}$ = 0
represents the instance at which the velocity vectors become detectable after
interaction in the downstream region. SD represents the standard deviation of
the data.

Figure 28: Figure S7: Backward FTLE field for (a) ${\phi_{1}\theta^{180}}$ and
(b) ${\phi_{2}\theta^{180}}$ shown at different $T^{*}$ corresponding to
figure 14 (in the main manuscript).