12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

ON LATE STAGES OF TRANSITION IN ROUND JETS

Naveen Balakrishna
Dept. of Aerospace Engg.

Joseph Mathew
Dept. of Aerospace Engg.

Arnab Samanta
Dept. of Aerospace Engg.

Indian Institute of Science
Bengaluru, India, 560012
naveenb@iisc.ac.in

Indian Institute of Science
Bengaluru, India, 560012
joseph@iisc.ac.in

Indian Institute of Technology
Kanpur, India, 208016
asamanta@iitk.ac.in

ABSTRACTsecondary structures, referred to in the literature as “halo vorticity”and “inner core”, which eventually break down to initiallyform hairpin vortices and later other rings or ring-like structures, thereby completing the transition to turbulence (Archer
et al., 2008). Here, the number of azimuthal waves around thering circumference determines the number of halo and hairpin
vortices.
The formation of ring-like vortices in the late stages oftransition from hairpin vortices were studied in the past using a
localized induction approach and the Biot-Savart law by Hama
(1962) and Moin et al. (1986), respectively. Both these studies included an isolated hairpin, modelled as a parabola, which
due to its tip curvature has larger normal self-induced velocities
there, thereby bringing in its legs closer together upstream of
the tip, beyond which point these studies were stopped owing
to worsening numerical accuracies. Additionally, Moin et al.
(1986) also studied the Biot-Savart evolution of this isolated
hairpin in the presence of a wall and uniform shear, and speculated that viscous vortex reconnection can occur when hairpin
legs come closer, leading to the formation of ring-like vortices.
However, such a vortex reconnection in hairpin vortices formed
during a round jet transition is yet to be demonstrated, which
we show in this work.
More recently, Biot-Savart evolution of antiparallel vortex rings has been studied, first in an ideal fluid by De Waele& Aarts (1994), who concluded that vortices form a pyramid
structure at their closest approach point, independent of initial conditions. Hussain & Duraisamy (2011) in their DNSof antiparallel line vortices included the effect of viscosity at
the closest approach point over a range of Reynolds numbers
𝑅𝑒 = Γ/𝜈 (250-9000), where Γ and 𝜈 are circulation and kinematic viscosity, respectively. They observed a cascade of viscous vortex reconnection events, confirming the conclusions ofMelander & Hussain (1988) that the viscous reconnection is notalways completed at one go and that there are left over vortices
that may undergo further reconnections at the smaller scales(see also Yao & Hussain, 2020b). Based on such observations,Moffatt & Kimura (2019a,b) proposed the pyramid model ofviscous vortex reconnection, where two approaching vortices
are set on two opposite faces of a pyramid that reconnect near
the pyramid tip, thereafter moving downwards to complementary faces, while the surviving vortices move upwards to thetip for further reconnections.
In this work, to demonstrate the viscous vortex reconnections in hairpin structures, we use the isolated hairpin configuration of Moin et al. (1986), but allow for its viscous evolution
via a DNS. Instead of a parabola, as taken by Moin et al. (1986),
our isolated hairpin is a semi-ellipse with Gaussian vorticity
distribution inside its core. Further, we also investigate the ef-

The bounding shear layer present near the nozzle exit of
low-speed round jets undergoes Kelvin-Helmholtz instability
that rolls up into vortex rings. These rings experience azimuthal instability and the ensuing nonlinear evolution sees theformation of secondary structures and the ejection of hairpin
vortices into the wake of the ring. The importance of hairpin
vortices in the laminar-turbulent transition process of round jets
has received little attention, and this is a preliminary work inthat direction. Using Direct Numerical Simulations (DNS), we
study the evolution of an isolated hairpin vortex, modelled as asemi-ellipse with a Gaussian vorticity distribution in its core.
The evolution is also explored in the presence of a uniformly
convecting stream and a uniformly sheared background flow.
The tip of the hairpin vortex moves up due to self-induction,
while its legs approach each other upstream of the tip. This
leads to a viscous vortex reconnection process leading to thepinch-off of ring-like vortices. Such reconnections of hairpin
vortices, considered here, are qualitatively similar to those observed in other configurations, like antiparallel line vortices.
As the Reynolds number increases, the time to reconnect decreases, while it occurs for increasingly smaller times. The
presence of a uniformly convecting background flow has noeffect on the reconnection process, while a uniform shear either accelerates or decelerates the process, depending upon thesense of this shear. This study shows hairpin vortex reconnection to be an important mechanism in the formation of small
scales in the wake of vortex rings during a round jet transition.

INTRODUCTION
Round jets, common in many industrial applications andnatural processes, are classified into laminar, transitional or
turbulent, based on the nature of nozzle-exit boundary layer.
An initially laminar jet usually transitions into a turbulent jet,which often involves multiple stages of several instability mechanisms. For example, the shear layer of a laminar jet is subjected to the classical Kelvin-Helmholtz instability (Michalke,
1984), which eventually rolls up into vortex rings via nonlinear
mechanisms. Such vortex rings are subjected to two types ofshort-wavelength azimuthal instabilities: elliptic (Widnall &Tsai, 1977) and curvature (Fukumoto & Hattori, 2005), withthe former being more commonly observed for rings duringtransition of round jets, which we study here. Elliptic instability leads to the formation of periodic waves around thecircumference of vortex rings, whose period depends upon theratio of core to ring radius and also on the details of vorticity
distribution inside the core (Balakrishna et al., 2020). Nonlinear evolution of these waves lead to the formation of complex

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Cases used for the study of hairpin vortex evolution.
The core radius is 0.3 for all the cases in this table.

fect of uniformly convecting background streams and uniformbackground shears on this hairpin, which act here as simplified
models for the presence of other vortical structures during around jet transition process.

NUMERICAL METHODS
The initial velocity field is obtained from a Gaussian vorticity field by solving Poisson equations, with the latter as thesource term. This vorticity field is specified inside the core ofan ellipse of radius 𝑎, which is centrally-located on the 𝑦 = 0
plane with an aspect ratio of AR = 𝑎 ′ /𝑏 ′ = 20, where 𝑎 ′ and 𝑏 ′are the lengths of the semi-major and semi-minor axis, respectively (see Fig. 1). The computational domain has dimensionsof 0 < 𝑥 < 𝐿 𝑥 , −𝐿 𝑦 /4 < 𝑦 < 3𝐿 𝑦 /4 and −𝐿 𝑧 /2 < 𝑧 < 𝐿 𝑧 /2, asshown in Fig. 1, with the length and time scales for the nondimensionalization being 𝑏 ′ and 𝑏 ′2 /Γ, respectively. In thiswork, both the DNS and the solutions to Poisson equations are
obtained via the open-source, incompressible Navier-Stokes
solver, Incompact3D (Laizet & Lamballais, 2009; Laizet & Li,2011).
In addition to the initial velocity field of the hairpin, we
also study cases where a uniformly convecting background flow
and a uniform background shear are added, to understand theirroles in the hairpin evolution. A free-slip boundary condition is
specified in the 𝑦 direction when simulating these backgroundflows, which are defined as 𝑢 𝑥 (𝑥, 𝑦, 𝑧) = 𝑆𝑦 +𝑈𝑐 , with 𝑈𝑐 chosen based on the slope 𝑆 such that the streamwise velocity
𝑢 𝑥 is non-zero at the lower boundary 𝑦 = −𝐿 𝑦 /4 (see Table 1
and Fig. 1). A Neumann condition is imposed at the 𝑥 = 0
boundary by forcing the normal derivative of all the velocity
components to zero, since it then allows the hairpin to movefreely at the inflow end, which gets distorted otherwise due toits self-induced velocity and the background flow, if present.
At 𝑥 = 𝐿 𝑥 , convective boundary conditions are applied, whilein the 𝑧 direction, we use periodic boundary conditions. The
length (same as that of Moin et al., 1986) and position of thehairpin are chosen such that its tip evolution is unaffected by
the boundaries. Further, an averaging operation is performedat each time step to preserve its initial symmetry about the 𝑧 = 0
plane (see Fig. 1).
The grid resolution is chosen based on the conservation of
volume-integrated kinetic energy (𝐸 𝑉 ) and the convergence of
𝐸 𝑉 and enstrophy (Ω). The maximum relative error between
the left and right-hand sides of the kinetic energy conservation
equation is less than 1.7% for the cases in Table 1, while
the relative error for enstrophy and volume-integrated kinetic
energy between the coarse (grid resolution of Table 1) and finegrid is ≈ 0.7% and ≈ 0.3%, respectively (see Naveen, 2022, forconvergence studies).

Case

𝐿𝑥

𝐿𝑦

𝐿𝑧

𝑁𝑥

𝑁𝑦

𝑁𝑧

𝑈𝑐

𝑆

𝑅𝑒

HW 1

25

20

10

641

513

256

0

0

3000

HW 2

25

20

10

641

513

256

0

0

1500

HU 1

35

20

10

897

513

256

0.2

0

1500

HU 2

35

20

10

897

513

256

0.1

0

1500

HS 1

35

20

10

897

513

256

0.2

0.01

1500

HS 2

35

20

10

897

513

256

0.2

-0.01

1500

decoding such a breakdown, we study the evolution of an isolated hairpin vortex, also in the presence of either a uniformly
convecting stream or a uniformly shearing flow. Using a quiescent flow, we first explore the effect of Reynolds number on thehairpin vortex evolution, followed by the background flows.
Our simulations with quiescent flow are designated with theprefix HW , while the cases with a uniformly convecting stream
and uniform shear have prefixes HU and HS , respectively (seealso Table 1).

Effects of Reynolds number
The effects of Reynolds number on the hairpin evolution is
explored in cases HW 1 and HW 2 of Table 1 and demonstrated
in Figs. 3-5. In both the cases, the initial evolution of thehairpin tip is similar to the Biot-Savart evolution of Moin et al.
(1986), where the entire hairpin shows an upward motion due
to its self-induced velocity as its legs act as counter-rotating
line vortices (compare the location of hairpin in (𝑎) and (𝑑) of
Fig. 3). Additionally, the hairpin tip is raised above the planeof its legs due to an increased self-induced velocity caused by
the small radius of curvature there. Later, with an increased
radius of curvature it attains a circular shape, which brings thehairpin legs closer upstream (see Fig. 3𝑏). The evolution of our
hairpin beyond this point departs from the Biot-Savart evolutionof hairpin line vortices, as studied by Moin et al. (1986), whowere unable to continue their Biot-Savart evolution further due
to worsening numerical accuracy. In spite of this, Moin et al.
(1986) had speculated the occurrence of vortex reconnection as
the viscous effects would start to dominate, splitting the initial
hairpin into a secondary hairpin and vortex ring. Our DNS
simulations clearly demonstrate this event of viscous vortex
reconnection and the associated splitting of the initial hairpin
vortex (see Figs. 3𝑏-𝑑).
In Fig. 4, a closer look of the reconnection process shows
it to be almost identical to that in other configurations, e.g. antiparallel line vortices (see Yao & Hussain, 2020b), where theself-induced velocity brings the hairpin legs closer, upstreamof its tip, leading to their collision. At the collision point, thevorticity in the adjacent legs gets annihilated due to viscosity
which leads to the closest vortex lines from the legs to get cut
and connect. Such connected vortex lines are next brought tothe top, get stretched due to the induced velocities, eventually
becoming perpendicular to the legs (see Melander & Hussain,1988). With time, more such vortex lines are connected andbrought to the top, thereby forming a perpendicular bridge(see Fig. 4𝑎). This process continues until the bridges start tomove away from each other due to their self-induced velocities,
leaving the unreconnected vortex lines in the form of antiparallel sheet-like structures, called threads (see Fig. 4𝑏). The

RESULTSWe first show the temporal evolution of azimuthal instability in an isolated vortex ring to demonstrate the occurrenceof halo and hairpin-like vortices. This is shown in Fig. 2 at
a few representative times, computed via Incompact3D following the procedure of Archer et al. (2008). Clearly, theformation of “halo vorticity”and their break down into a series of hairpin-like vortices can be seen in Figs. 2(𝑏) and (𝑐),respectively (see Naveen, 2022, for more details). Note that unlike in wall-bounded flows, here, such hairpin-like structures
seem to coexist with a range of smaller scales so the latter may
be presumed to have a role on the evolution of these hairpins,
leading to the eventual ring breakdown. As a first step toward2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
3𝐿 𝑦 /4(𝑎)𝐿 𝑧 /2

(𝑏)

𝑦

𝑧 0

0
−2

0

2

−𝐿 𝑧 /20

𝐿𝑥𝑥−𝐿 𝑦 /4−𝐿 𝑧 /20𝑧

𝐿 𝑧 /2

Figure 1. (𝑎) Plan view and (𝑏) end view showing the initial location of the hairpin vortex in the computational domain visualised with
contour of 𝜆 2 = −0.1 and coloured with axial vorticity (𝜔 𝑥 ). The 𝑧 = 0 and 𝑦 = 0 planes are represented as · · · · · ·and · · · · · ·, respectively.
The grey arrows in (𝑏) indicate the background velocity field (𝑢 𝑥 ) for the cases HU 1 (𝑧 < 0) and HS 1 (𝑧 > 0).

(𝑏)

(𝑎)

(𝑐)

Hairpin-like vortices

(a

(c)

Halo
vor
Halo
vorticity(

Hairpin-like vortices

Figure 2. Evolution of vortex ring shown with contours of 𝜆 2 = −0.001 at different times (a) t = 110; (b) t = 150; (c) t = 176.5.

(𝑎)

(𝑏)
Bridge500Thread

20

(𝑐)

(𝑑)
5
0
𝑦
𝑥
𝑧

Vortexdipole0Secondary
hairpinvortex−0.531

Vortex
ring
0

20

0.531

Figure 3. Isosurfaces of vorticity magnitude |𝝎| = 0.15|𝝎| 0 coloured with contours of 𝜔 𝑥 for the case HW 1 at 𝑡 = 0 (𝑎), 𝑡 = 55 (𝑏),𝑡 = 65 (𝑐) and 𝑡 = 80 (𝑑). The inflow plane is coloured in grey and a part of axis is provided in (𝑎) and (𝑑) to show the upward motionof the hairpin vortex as the simulation proceeds from 𝑡 = 0 (𝑎) to 𝑡 = 80 (𝑑). Here, |𝝎| 0 is the maximum vorticity magnitude at 𝑡 = 0.
induced velocity of the bridges between the threads is similar
to a planar jet-like flow, which reverses the curvature of thethreads and slows down their reconnection. At the same time,the mutual induction between these threads causes their peak3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Head-tailstructure
VortexBridge(𝑏)(𝑐)(𝑑) dipole

(𝑎)

𝑦

𝑥

𝑧
−0.531ThreadFigure 4.
= 80.

0

0.531

A closer look at the viscous vortex reconnection process in Fig. 3 at different times (a) t = 55; (b) t = 57; (c) t = 65 and (d) t

(𝑎)

(𝑏)
150

1.5

100

1
𝑑

Ω𝑚
50

0.5
0
0

20

40

𝑡

60

0

80

0

20

40

60

𝑡

80

······

Figure 5. Time evolution of (𝑎) separation distance and (𝑏) maximum enstrophy for the cases HW 1 () and HW 2(). The vertical dashed lines denote minimum separation distance at 𝑡 = 53 (HW 1) and 𝑡 = 61 (HW 2) in (𝑎), while in (𝑏) they
denote local peaks at 𝑡 = 57 (HW 1) and 𝑡 = 65 (HW 2).

—note that the theoretical model of Moffatt & Kimura (2019b)
predicted the presence of a physical, but not a mathematicalsingularity, while the DNS results of Yao & Hussain (2020a)
showed that even a physical singularity is impossible, eventhough its occurrence in the inviscid limit is unknown yet.
As expected, increased viscosity also leads to thicker vortex cores and slower movements of the reconnection plane
towards the inflow (see for case HW 2), but the three stages ofreconnection identified by Melander & Hussain (1988) remains
identical. For even higher Reynolds numbers (𝑅𝑒 > 3000), thebridges move away faster, leading to threads that cut and reconnect, while the tail portion of the head-tail structure become
unstable via Kelvin-Helmholtz instability, rolling up into multiple dipoles. These head dipoles undergo further reconnections
leading to the formation of vortex rings, hairpin packets andseveral small-scale structures (Yao & Hussain, 2020b; Mcgavin& Pontin, 2018; Beardsell et al., 2016; Hussain & Duraisamy,2011).

vorticity regions to convect faster, leading to the formation ofa head-tail structure (see Fig. 4𝑐), where the head part forms
a vortex dipole, while the tail gets dissipated due to viscosity(see Fig. 4𝑑). However, unlike antiparallel line vortices, incases HW 1 and HW 2, the reconnection plane actually moves
towards the inflow plane due to the self-induced velocity of thehairpin tip (see Figs. 3𝑎 and 3𝑑). In fact, Melander & Hussain(1988) divided this whole process into three stages: (I) inviscid
advection, (II) bridging and (III) threading, all of which areclearly visible in Figs. 3 and 4.
The evolution of the hairpin at 𝑅𝑒 = 1500 (case HW 2 ofTable 1) is identical to that of the higher Reynolds number
case HW 1, except the onset of the reconnection process getting
delayed and its duration extended, which can be observed fromthe evolution of separation distance 𝑑 and maximum enstrophy
Ω𝑚 , shown in Fig. 5. For finding 𝑑, the vortex centre tracking
method using vortex lines is used (see Yao & Hussain, 2020c),
where a single vortex line is injected from the vorticity centroidof the hairpin leg at 𝑧 > 0, such that it traces the entire hairpin
vortex. This yields the minimum distance between the hairpin
legs before reconnection, while after reconnection two vortex
lines are needed to find the separation distance: one of which is
injected from the inflow plane and the other from the vorticity
centroid of the vortex ring core in 𝑧 = 0 plane.

Effects of background flows
In Figs. 6 and 7, we compare the cases with backgroundflows with the case in quiescent fluid, all at 𝑅𝑒 = 1500, via
the changes in different parameters due to the inclusion of
background flows. The isosurfaces of vorticity magnitude in
Fig. 6 clearly show the hairpin vortex to get convected alongwith the corresponding background flow, increasing the lengthsof hairpin legs, more so for the case with a higher convection
velocity (see case HU 1 of Fig. 6𝑎), as expected. Similarly,
the case with positive shear (case HS 1 of Fig. 6𝑐) convects
the hairpin more than that with negative shear (case HS 2 ofFig. 6𝑑), as in the former the hairpin moves into higher velocity
regions of the background flow as time progresses. In general,the background stream opposes the self-induced motions of thehairpin tip, carrying the tip along with it. Note that Fig. 6 also
shows the onset of a second reconnection process upstreamof the first, indicating the occurrence of a sequence of such

At the lower 𝑅𝑒 (case HW 2), the reconnection is delayed
and thus both the separation distance attaining its minimum(see Fig. 5𝑎) and the maximum enstrophy reaching its local
peak happen at later times (see Fig. 5𝑏). Note that the valueof Ω𝑚 peaks after reconnection due to the formation of smallscale structures (Hussain & Duraisamy, 2011; Yao & Hussain,
2020b), while in Fig. 5(𝑏), multiple peaks in Ω𝑚 at 𝑅𝑒 = 3000
are visible, whereas for 𝑅𝑒 = 1500, a single peak is seen. In
fact, the number of such peaks and their magnitudes increases
with 𝑅𝑒, which is important in determining whether a finite
time singularity (FTS) is possible for the Navier-Stokes equations (see Beale et al., 1984; Fefferman, 2006). We briefly

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(𝑎)

(𝑏)

5

5
0

0
0

0
Secondreconnection

30

30

(𝑐)

(𝑑)

5

5
0

0
𝑦

0
𝑧

0
𝑥

−0.531

30

0

0.531

30

Figure 6. Isosurfaces of vorticity magnitude |𝝎| = 0.15|𝝎| 0 coloured with contours of 𝜔 𝑥 for the cases HU 1 (𝑎), HU 2 (𝑏), HS 1 (𝑐)and HS 2 (𝑑) at 𝑡 = 85. The rest are same as figure 3.

(𝑎)

(𝑏)
40

2
Ω𝑚

𝑑

20

1

0

0
0

20

40

𝑡

60

80

0

20

40

𝑡

60

80

Figure 7. Time evolution of (𝑎) separation distance and (𝑏) maximum enstrophy for the cases HW 2 (•), HU 1 (×), HU 2 (+), HS 1). The minimum 𝑑 occurs at 𝑡 = 60.5 (HU 1, HU 2), 𝑡 = 61 (HW 2, HS 2) and 𝑡 = 61.5 (HS 1), while the local() and HS 2 (peak in Ω𝑚 occur at 𝑡 = 65 (HW 2, HU 1, HU 2, HS 2) and 𝑡 = 65.5 (HS 1).

——

reconnections as the hairpin is convected by the backgroundflow.

of the process for cases HS 1 and HS 2, when compared to theuniformly convecting cases. On comparing the shear cases, thelocal peak of Ω𝑚 and the minimum of 𝑑 occurs later for HS 1,
indicating a delayed onset of reconnection, while for HS 2, theyoccur earlier at the same time as that of HW 2. At this stage,
the reason for such differences is not clear and more simulations with different background flows and Reynolds numbers
are being carried out, whose findings will be reported in the
future.

As shown in Fig. 7𝑎, the time evolution of separation distances for all the cases at 𝑅𝑒 = 1500 studied here (including thequiescent case HW 2) are largely similar before the reconnection event, except for the slight earlier occurence of the distanceminimum for the uniformly convecting flow cases of HU 1 andHU 2. Further, uniform convection yields an insignificant effect on the evolution of Ω𝑚 and its peak value (see Fig. 7𝑏).
Note that to compare with the HW 2 case, the corresponding
background flows are first subtracted before calculating enstrophy for the other cases. Hence, it may be inferred that theuniform convection of background stream does not alter thereconnection process in any way, except for the convection ofreconnection plane downstream, thereby reconfirming the local
nature of this process as is also reported in the literature.

SUMMARY
The main findings of this work are summarised in thefollowing.
1. The evolution of an isolated hairpin vortex in a quiescent fluid is identical to the Biot-Savart evolution of thevortex line till pinch-off (see Moin et al., 1986). The initial hairpin gets split into a secondary hairpin and vortex
ring due to the viscous vortex reconnection process, as
observed with the present hairpin configuration, which isalso identical to other configurations reported in the literature, especially antiparallel line vortices (Yao & Hussain,
2020b).
2. A decrease in Reynolds number leads to a delay in theonset of reconnection and an increase in the duration ofthis process.

In contrast to the uniformly convecting flow cases, once
uniform background shear is applied, although it still has arather small effect on 𝑑 during the initial vorticity rearrangement phases, but such a flow significantly alters the dynamics
post reconnection, as is visible in Fig. 7(𝑎). This different
evolution after reconnection occurs as different parts of vortex structures, especially the vortex rings, experience different
background flow velocities, manifested via varying distances
between the secondary hairpin and vortex ring in Fig. 6. The
corresponding reconnection time obtained by fitting the linear
portion of 𝑑 before reconnection indicated slight delayed onset

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Laizet, S. & Lamballais, E. 2009 High-order compact schemes
for incompressible flows: A simple and efficient methodwith quasi-spectral accuracy. J. Comput. Phys. 228 (16),5989–6015.
Laizet, S. & Li, N. 2011 Incompact3d: A powerful tool to tackle
turbulence problems with up to o (105) computational cores.
Int. J. Numer. Methods Fluids 67 (11), 1735–1757.
Mcgavin, P. & Pontin, D. I. 2018 Vortex line topology during
vortex tube reconnection. Phys. Rev. Fluids 3 (5), 054701.
Melander, M. V. & Hussain, F. 1988 Cut-and-connect of two
antiparallel vortex tubes. In Studying Turbulence Using Numerical Simulation Databases, , vol. 2, pp. 257–286. Center
for Turbulence Research.
Michalke, A. 1984 Survey on jet instability theory. Progress in
Aerospace Sciences 21, 159–199.
Moffatt, H. K. & Kimura, Y. 2019a Towards a finite-time singularity of the navier–stokes equations part 1. derivation andanalysis of dynamical system. J. Fluid Mech. 861, 930–967.
Moffatt, H. K. & Kimura, Y. 2019b Towards a finite-time singularity of the navier–stokes equations. part 2. vortex reconnection and singularity evasion. J. Fluid Mech. 870.
Moin, P., Leonard, A. & Kim, J. 1986 Evolution of a curved
vortex filament into a vortex ring. Phys. Fluids 29 (4), 955–
963.
Naveen, B. 2022 On late stages of transition in round jets. PhD
thesis, Indian Institute of Science.
Widnall, S. E. & Tsai, C. Y. 1977 The instability of the thinvortex ring of constant vorticity. Philos. Trans. Royal Soc. A
287 (1344), 273–305.
Yao, J. & Hussain, F. 2020a On singularity formation via viscous vortex reconnection. J. Fluid Mech. 888.
Yao, J. & Hussain, F. 2020b A physical model of turbulence
cascade via vortex reconnection sequence and avalanche.
J. Fluid Mech. 883.
Yao, J. & Hussain, F. 2020c Separation scaling for viscous
vortex reconnection. J. Fluid Mech. 900, R4.

3. The addition of a background flow leads to the increasein length of hairpin legs due to the convection of hairpin
tip with the flow. The uniformly convecting flows do notalter the reconnection process. In contrast, positive andnegative shear accelerates and decelerates, respectively,
the onset of reconnection.
Partial funding from the Office of Naval Research Global(ONRG) NICOP grant with award number N62909-17-1-2131
with Dr. S. Ahmed as the Project Monitor is gratefully acknowledged. For computing resources, we acknowledge the use of
the “SahasraT” Cray XC40 system at the Supercomputer Education and Research Centre (SERC) of the Indian Institute of
Science for most of our simulations.

REFERENCES
Archer, P. J., Thomas, T. G. & Coleman, G. N. 2008 Direct numerical simulation of vortex ring evolution from the laminarto the early turbulent regime. J. Fluid Mech. 598, 201–226.
Balakrishna, N., Mathew, J. & Samanta, A. 2020 Inviscid andviscous global stability of vortex rings. J. Fluid Mech. 902.
Beale, J. T., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3-d Euler equations. Commun. Math. Phys. 94 (1), 61–66.
Beardsell, G., Dufresne, L. & Dumas, G. 2016 Investigation of
the viscous reconnection phenomenon of two vortex tubes
through spectral simulations. Phys. Fluids 28 (9), 095103.
De Waele, A. T. A. M. & Aarts, R. G. K. M. 1994 Route tovortex reconnection. Phys. Rev. Lett. 72 (4), 482.
Fefferman, C. L. 2006 Existence and smoothness of the NavierStokes equation. In The Millennium Prize Problems, pp.
57–70. American Mathematical Society.
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of avortex ring. J. Fluid Mech. 526, 77–115.
Hama, F. R. 1962 Progressive deformation of a curved vortex
filament by its own induction. Phys. Fluids 5 (10), 1156–
1162.
Hussain, F. & Duraisamy, K. 2011 Mechanics of viscous vortex
reconnection. Phys. Fluids 23 (2), 021701.

6