# A DNS Study of entrainment in an axisymmetric turbulent jet as an episodic
process

Prasanth Prabhakaran<EMAIL_ADDRESS>- presently at Michigan Technological
University Max Planck Institute for Dynamics and Self-Organization,
Goettingen, 37077, Germany
Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific
Research, Bangalore 560064, India  Sachin Y. Shinde<EMAIL_ADDRESS>Department of Mechanical Engineering, Indian Institute of Technology, Kanpur,
208016,, India
Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific
Research, Bangalore 560064, India  Roddam Narasimha<EMAIL_ADDRESS>Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific
Research, Bangalore 560064, India

###### Abstract

This investigation is based on a Direct Numerical Simulation (DNS) of a steady
self-preserving incompressible axisymmetric turbulent jet at a Reynolds number
of $2400$ (based on orifice diameter $d_{0}$ and mean exit velocity
$\overline{w}_{0}$ at floor level $z~{}=~{}0$). The DNS data enable accurate
maps of the outer irrotational flow field, and also the vorticity field
$\mathbold{\omega({x}},t)$ in the turbulent core of the jet. It is found
necessary to define two separate boundaries of the jet. The first is an inner
boundary (turbulent/nonturbulent, T/NT) at $|\mathbold{\omega(x,}t)|=0.5$
(quoted in units of local center-line velocity and half-width), from where
$|\mathbold{\omega(x},t)|$ rises steeply towards the turbulent core whose
boundary is located around $|\mathbold{\omega(x},t)|=1.0$. The second is an
outer rotational/irrotational (R/IR) boundary at
$|\mathbold{\omega(x},t)|=0.1$, beyond which the flow may be considered
irrotational. In the diametral and axial sections, the separation between the
inner and outer boundaries can vary from $2\eta$ to $5\lambda$ ($\eta$ and
$\lambda$ are the Kolmogorov and Taylor length scales respectively). In the
latter case, $|\mathbold{\omega(x},t)|>0.1$ between the two boundaries but is
not stochastic, so the inter-boundary region is in the nature of a viscous
laminar sheath. This region can often be traced to a fossil of a vortical
tongue that has a long life of the order of a few hundred flow units
($d_{0}/\overline{w}_{0}$), because of the very low advective velocity towards
the edge of the jet. The velocity field beyond the outer boundary often has
ordered, nearly irrotational circulatory motions. These can be shown, in
simpler cases, to be the velocity field induced by one or more vorticity
elements in a coherent structure in the turbulent core. A detailed examination
of axial and diametral sections indicates that there are periods when there is
a large inrush of ambient fluid into parts of the T/NT interface, which gets
distorted into a ‘gulf’ or ‘well’ that can be both twisted and deep. Sections
of these wells often appear as what may be called as ‘lakes’ of irrotational
fluid in diametral sections of the jet flow. Part of the inrushing fluid
crosses the T/NT interface within the well and is entrained into the turbulent
core, by a process that can legitimately be called nibbling. The duration of
such an inrush process can be of the order up to $20$ flow units and suggests
that entrainment can be an episodic process in which an inrush event
accelerates ambient fluid even as it is pushed into a narrowing gulf, where it
penetrates the T/NT interface of the gulf by nibbling. This supports the view
that engulfment and nibbling are successive stages in the life of an
entrainment episode. In view of the well-known difficulties associated with
the analysis of fluid flow past a highly convoluted 3D fractal interface, the
flow is analyzed here by the device of enclosing the jet in $z_{1}\leq z\leq
z_{2}$ by a minimal cylindrical disk of height $z_{2}-z_{1}$ and radius nearly
touching the radially farthest point on the inner edge of the jet anywhere
within the disk. By analyzing the radial flow at the edge of the disk with
flow at the T/NT interface in the same angular sector, it is found that the
two flows are correlated in about $90$% of the time and anti-correlated the
rest of the time. A mass flux budget across the surfaces of the disk and the
turbulent core of the jet enables determination of the inrush events around
the polar angle in the diametral section, and suggest an average of 12 to 16
events in $360^{\circ}$. This episodic view permits the introduction of such
flow parameters as burstiness, which is a measure of the compactness and
intensity of the events under study, as used in the analysis of momentum flux
events in the atmospheric boundary layer Narasimha  _et al._ (2007). In the
present case, similar episodes are responsible for entrainment. In the
turbulent round jet, the entrainment burstiness is found to be of order
$0.75$, comparable to the momentum flux burstiness found in a turbulent
boundary layer.

###### pacs:

47.32.ck, 47.63.M-, 47.27.wg

##  I Introduction

The process by which ambient fluid is entrained into a turbulent shear flow
has been a question of great importance for a long time. The first significant
study of the problem, due to Corrsin (1955) Corrsin and Kistler (1955), was
inspired by a striking instantaneous shadowgraph of the turbulent wake of a
bullet traveling at supersonic speeds. This picture showed a sharp boundary
between the turbulent core of the wake and the ambient fluid. Corrsin proposed
that entrainment occurred due to what has come to be called the ‘nibbling’ of
the ambient fluid at the turbulent / non-turbulent (T/NT) interface of the
flow across a viscous super-layer whose thickness scales with the Kolmogorov
length. On the other hand, the work of Brown & Roshko (1972, 1974) Brown and
Roshko (1972, 1974) presented striking optical visualizations of flow in
turbulent free-shear (or mixing) layers showing highly organized structures,
and concluded that entrainment occurred by ‘engulfment’ of ambient fluid
through the velocity induced by the coherent structures present in the
turbulent core flow. Studies by Dimotakis (1983) Dimotakis  _et al._ (1983) on
an axisymmetric jet showed that the instantaneous boundary of the turbulent
core (as marked by passive scalar dye concentration) was highly convoluted,
and supported the idea for a dominant role of coherent structures in
entrainment, by engulfment of ambient fluid.

The main issue in much recent work on entrainment in turbulent shear flows can
be summarized as a debate on nibbling vs. engulfment as primary candidates for
the entrainment mechanism. While it is realized that it is not easy to make a
precise distinction between the two (Da Silva (2014) da Silva  _et al._
(2014)), recent work, conceding that engulfment may be responsible for most of
the entrainment in mixing layers (Westerweel et al. (2002) Westerweel  _et
al._ (2002)), has generally concluded that it accounts for little in other
turbulent flows like jets, wakes (Bisset et al (2002) Bisset  _et al._ (2002))
and grid turbulence (Holzner & Luethi (2011) Holzner and Lüthi (2011)). A
related part of the debate has been concerned with determining the length and
velocity scales relevant to the entrainment process, and in particular the
roles played by the Kolmogorov and Taylor length scales therein Westerweel
_et al._ (2009); da Silva and Taveira (2010); da Silva  _et al._ (2014).

On these issues the jet has been the most widely studied turbulent shear flow
(experiments on round jets reported by Westerweel et al. (2002) Westerweel
_et al._ (2002), Wolf et al. (2012) Wolf  _et al._ (2012); DNS results on a
temporal round jet by Mathew & Basu (2002) Mathew and Basu (2002), on a
temporally evolving plane jet by Reeuwijk and Holzner (2014) van Reeuwijk and
Holzner (2014) and on a spatially developing plane jet by Da Silva (2010) da
Silva and Taveira (2010)). The analysis of Mathew & Basu Mathew and Basu
(2002) concludes that most of the entrainment occurs due to small-scale action
at the T/NT interface. This idea was further supported by Westerweel et al.
(2005, 2009) Westerweel  _et al._ (2005, 2009) and Wolf et al. (2012) Wolf
_et al._ (2012) based on PIV measurements on a round jet. All these studies
report that small-scale activity at the T/NT interface plays a dominant role
in the entrainment process. Westerweel et al. (2009) Westerweel  _et al._
(2009) concluded that large scale engulfment contributed about $8$% of the net
mass flux. Wolf et al. (2012) Wolf  _et al._ (2012) used particle tracking
velocimetry to generate the 3D velocity field and compute the interface
velocity using this data. Their studies suggested that the entrainment
velocity is dominated by viscous action due to the small scale eddies at the
T/NT interface. They also show that the entrainment velocity is strongly
influenced by the local curvature of the interface with higher entrainment at
the concave region of the T/NT interface. The analysis on the DNS of temporal
jets by Reeuwijk & Holzner (2014) van Reeuwijk and Holzner (2014) shows the
presence of a viscous super layer (VSL) enveloping the vortical core. They
also define a buffer region similar to those in wall bounded flows, which
connects the VSL with the turbulent core.

Philip and Marusic (2012) Philip and Marusic (2012) proposed a large eddy
model to reproduce some of the features of the large scale motion in jets and
wakes. Recent work by Mistry et al. (2016) Mistry  _et al._ (2016), using
simultaneous PIV and PLIF measurements, stresses the multi-scale nature of the
entrainment velocity and the scale-independence of mass flux in a turbulent
jet. These studies, along with the work of Turner (1986) Turner (1986), Dahm &
Dimotakis (1987) Dahm and Dimotakis (1987), Mathew & Basu (2002), Wolf et al.
(2012) Wolf  _et al._ (2012) and Reeuwijk & Holzner (2014) van Reeuwijk and
Holzner (2014) propose that entrainment dynamics is dictated or imposed by
large-scale coherent motions, to which small scale viscous action adjusts
itself to maintain a scale-independent mass flux rate.

In a recent work Burridge et al. (2017) Burridge  _et al._ (2017) have
reported simultaneous PIV/PLIF measurements on the entrainment dynamics in a
turbulent axisymmetric plume. They present conditional statistics highlighting
the role of large scale motions in the entrainment process. Their studies
suggest that significant momentum is imparted to the fluid outside the T/NT
interface in the zones between the coherent structures. Moroever, they
conclude that turbulent entrainment in a plume is dominated by engulfment in
the first stage followed by viscous nibbling. Similar conclusions on the role
of the large scale motions were reported earlier by Plourde et al. (2008)
Plourde  _et al._ (2008). They associate the presence of large scale organized
motions with higher instantaneous entrainment levels.

To the best of our knowledge there has been no numerical study of the round
jet from the entrainment point of view, and the present work was undertaken in
part to fill this gap. The big advantages of having DNS data are that the
vorticity vector field can be accurately determined, enabling a vorticity view
of the T/NT interface, and the flow field in the ambient fluid beyond the edge
of the jet can be accurately determined. The present report starts by
presenting, as a first step, a physical picture of entrainment largely derived
from DNS imagery. This is followed by an analysis of the entrainment budget
across a cylindrical disc of finite thickness circumscribing the jet in the
self-preserving zone of the jet. The analysis strongly suggests an episodic
(spatially discrete and temporally intermittent) nature of the entrainment
process. One reason for this approach is that while much attention has been
given in earlier studies to flow within and very close to the fully turbulent
core, surprisingly little attention has been given to the structure of the
flow in the ambient fluid outside the turbulent core in a slightly more
extended neighborhood (over an area with a length scale of order $b_{w}$, the
radius of the jet at half the mean center-line velocity). This may partly be
due to difficulties in making accurate measurements of the very low velocities
in the ambient fluid. Furthermore, early models for what may be called the
‘ambient near-field’ visualized it as resulting from a distribution of sinks
along the center-line of the jet (Phillips (1955) Phillips (1955), Bradshaw
(1976) Bradshaw (1976)), but this approach does not throw any light on the
spatio-temporal structure of the flow in the ambient fluid. A very useful
recent review of these developments has been presented by Da Silva et al.
(2014) da Silva  _et al._ (2014).

Several definitions of entrainment have been proposed in the literature
(Turner (1986) Turner (1986)). All the experiments and simulations on jets and
wakes discussed so far define entrainment as the velocity at which the
turbulent flow spreads into the irrotational ambient. Here we define
entrainment as the rate at which the irrotational ambient fluid enters into
the vortical turbulent core of the jet. The two definitions have significantly
different implications which are best understood by looking at a planar wake.
According to the former definition of entrainment, a planar wake has non-zero
entrainment as the wake spreads in the streamwise direction. On the contrary,
the wake has a net zero entrainment using the latter definition as the average
mass flux in the streamwise direction is a constant. It is important to
distinguish between encroachment (as in the spreading wake with no change in
mass flux) and entrainment (which involves an increase in turbulent mass
flux). In a turbulent round jet, the net mass flux in the streamwise direction
as well as the diameter increases downstream. We investigate here the
entrainment dynamics close to the T/NT interface in a turbulent round jet.

Against this background we present data on the instantaneous flow field
outside the turbulent core in a round jet, as obtained by a direct numerical
simulation of the jet flow. This makes it possible to visualize the general
structure of the ambient flow field, and examine its possible relation to the
vorticity field (or other flow characteristics) within the jet. A general
schematic view of the velocity field, as sketched for example by Philip &
Marusic (2012) Philip and Marusic (2012), implies that the ambient fluid flows
into the turbulent jet along more or less parallel streamlines almost right up
to the T/NT interface, where it is entrained into the jet core across its
edge. A somewhat more structured velocity field is indicated by the
measurements reported by Westerweel et al. (2009) Westerweel  _et al._ (2009)
(Figs. 19 and 20). Here we present evidence, chiefly through flow imagery,
indicating that the instantaneous flow in the ambient neighborhood of the
turbulent edge of the jet is much more organized than thought previously, and
the entrainment across the T/NT interface is episodic in nature. A more
detailed quantitative study will appear separately.

The paper is organized into ten sections. We start with the description of the
numerical procedure used for the simulation and the details on the accuracy of
the data, followed by a discussion on the identification of the boundaries
separating the turbulent core from the (nearly) irrotational ambient. Later,
the axial and diametral sections are analysed to understand the dynamics near
the T/NT interface. Based on these details we conduct a quantitative analysis
of the mass flux near the T/NT interface followed by concluding remarks.

##  II Simulation Details

Fig. 1 shows a schematic of an axial section of the jet, where $d_{0}$
represents the orifice diameter from which the jet issues with a top hat
velocity profile characterized by a uniform jet exit mean velocity
$\overline{w}_{0}$. The flow is governed by the incompressible Navier-Stokes
equations (in non-dimensional form with jet exit velocity scale
$\overline{w}_{0}$ and nozzle exit diameter $d_{0}$ as velocity and length
scales respectively),

$\nabla\cdot\bm{u}=0,$ (1)
$(\partial{\bm{u}}/\partial{t})+(\bm{u}\cdot\nabla)\bm{u}=-\nabla{p}+(1/\mbox{{Re}})\nabla^{2}\bm{u},$
(2)

where $\bm{u}$, $p$ represent the flow velocity vector and pressure
respectively, and $\mbox{{Re}}=\overline{w}_{0}d_{0}/\nu$ is the Reynolds
number (= 2400 in the present simulations). Unless otherwise specified, all
distances, velocities and time presented here are non-dimensionalized with
$d_{0}$ and $w_{0}$ as scales, and vorticity with the time-mean local scales
at height $z$, viz., centerline velocity $\overline{w}_{c}(z)$ and half-
velocity jet width $b_{w}(z)$. Also note that, unless otherwise stated, all
flow variables are non-dimensional.

\begin{overpic}[width=227.62204pt,height=264.6107pt,unit=1mm]{Fig_box.eps}
\put(0.0,0.0){\includegraphics[width=227.62204pt]{Fig_1.eps}} \end{overpic}
Figure 1: Diagram of an axial section to illustrate main terminology and
notation for jet flow. (See additional explanation in text.)

The equations are solved using the extension of Harlow & Welsh’s scheme Harlow
_et al._ (1965) to non-uniform grids as proposed by Verstappen & Veldman
(2003) Verstappen and Veldman (2003), using a second-order finite volume
framework. The governing equations are integrated in time using the second-
order Adams-Bashforth scheme with a time step $\Delta t=0.005$. A non-uniform
Cartesian mesh is used to resolve all the relevant scales in the turbulent
part of the flow-field. The velocity and pressure variables are stored in a
staggered arrangement to prevent pressure-velocity decoupling. The pressure
variable is stored in the cell center and the velocity components are stored
on the cell faces Harlow  _et al._ (1965). More details on the scheme and the
code are discussed in Prasanth (2014) Prasanth (2014). The computational
domain extends up to $z=55$ in the streamwise ($z$) direction and $-50$ to
$+50$ in the cross-stream directions ($x,y$). The total grid size is
$480\times 480\times 600$ in the $x$, $y$ and $z$ directions respectively
(yielding 138 million grid points in all). In the streamwise direction the
grid size varies from 0.05 at the orifice to 0.18 at the outflow. In the
cross-stream plane the grid is uniform (of size 0.05) from $x$, $y$ = $-9.25$
to $+9.25$, thus ensuring adequate resolution in the turbulent core; beyond
this the grid is gradually stretched towards the lateral side walls. The mean
viscous dissipation $\overline{\epsilon}$ is calculated from the fluctuating
strain rate $s_{ij}^{\prime}$ as

$\overline{\epsilon}=2\nu\overline{s_{ij}^{\prime}s_{ij}^{\prime}},$ (3)
$s_{ij}^{\prime}=\frac{\partial{u_{i}^{\prime}}}{\partial
x_{j}}+\frac{\partial{u_{j}^{\prime}}}{\partial x_{i}},$ (4)

where the primes denote fluctuating quantities
($u_{i}^{\prime}=u_{i}-\overline{u}_{i}$, etc.) The velocity derivatives are
calculated using a second order central differencing scheme. The Kolmogorov
and Taylor microscales are respectively given by

$\eta={\nu}^{3/4}\,{\overline{\epsilon}}^{-1/4},$ (5)
$\lambda=(15\nu\overline{{w^{\prime}}^{2}}/{\overline{\epsilon}})^{1/2},$ (6)

where $\overline{{w^{\prime}}^{2}}$ is the turbulence intensity in the
streamwise direction. The dissipation (normalized using $w_{c}^{3}/(z-z_{0})$,
where $z_{0}$ is a virtual origin) near the jet centerline at $z\approx 33$ in
the present simulation is $0.18$, to be compared with $0.18$ reported by
Panchapakeshan & Lumley Panchapakesan and Lumley (1993) at $Re=11,000$, $0.22$
by Bogey & Bailly (2009) Bogey and Bailly (2009) at $Re=11,000$ and $0.25$ by
Taub et al. (2013) at $Re=2000$ Taub  _et al._ (2013). At $z/d\approx 33$ near
the jet center-line, $\lambda/b_{w}\simeq 0.2$, to be compared with 0.18
(Westerweel  _et al._ (2009), round jet, $\mbox{{Re}}=2000$, center-line). The
Kolmogorov length scale ($\eta$) at $z\approx 33$ is about $0.04~{}d_{0}$, and
the corresponding grid size in the $x$, $y$, $z$ directions is approximately
$\eta$, $\eta$ and $3\eta$ respectively.

The bottom plane at $z=0$ outside the orifice is treated as a no-slip wall in
the plane of the orifice exit. A fixed, uniformly distributed disturbance of
5% amplitude with zero mean is superposed on the top hat velocity profile at
the orifice to trip the flow to a turbulent state. The disturbance is
superposed only on the streamwise velocity component, however, and at a random
set of points also chosen from a uniform distribution over the orifice area.
At the outflow boundary we use the zero normal derivative condition for all
the variables, with a layer of viscous padding from $z=52$ to $z=55$ to ensure
smooth exit of the turbulent flow from the computational domain. The lateral
boundaries of the computational domain are also treated as no slip walls, but
(at $x,y=\pm 50$) they are sufficiently far away from the jet axis to have any
significant effect on the momentum balance. The net momentum flux
$\int(\overline{w}^{2}+\overline{{w^{\prime}}^{2}}-0.5[\overline{{u^{\prime}}^{2}}+\overline{{v^{\prime}}^{2}}])dS$,
where the overbar denotes time average, prime denotes fluctuation and the
domain of integration is $-50\leq x,y\leq 50$, is conserved to better than 99%
over the whole domain in the $z$ direction (Shinde et al. (2019) Shinde  _et
al._ (2019)). Other details about the solver and its validation are discussed
in Prasanth (2014) Prasanth (2014). The simulation took up to $t=900$ to
attain a stationary state. Data for computing flow statistics was acquired
over the time interval $900<t<2650$. The simulation was run on 108 nodes of
the 360 TF supercomputer at CSIR-4PI, Bangalore.

##  III Data analysis

The mean streamwise component of the velocity at $r,z$ is computed as

$\lim_{t\rightarrow\infty}\frac{1}{t}\int_{t_{1}}^{t_{2}}\frac{1}{2\pi}\int_{0}^{2\pi}\bm{w}(r,\bm{z},\phi,t^{\prime})d\phi
dt^{\prime}=\overline{\bm{w}}(\bm{r,z}),$ (7)

where $t_{1}=900$, $t_{2}=2600$, $t=t_{2}-t_{1}$, and $\phi$ is the azimuthal
angle. It is found that the jet is in a self-similar state over the range
$27\leqslant z\leqslant 40$ to within $0.6\%$ in the mean velocity
($\overline{w}/\overline{w}_{c}$) and in an equilibrium or self-preserving
state over the range $32\leqslant z\leqslant 36$, the maximum in
$(\overline{{w^{\prime}}^{2}}/\overline{w}^{2}_{c})^{0.5}$ and Reynolds shear
stress $-\overline{w^{\prime}u^{\prime}}/\overline{w}^{2}_{c}$ vary less than
$1\%$ and $1.5\%$ respectively. Additional details on the statistics are
available in Shinde et al. (2019) Shinde  _et al._ (2019).

We present here analyses of the solutions for the vorticity field within the
turbulent jet and the velocity field outside a suitably determined ‘edge’ of
the turbulent core of the jet, defined as the contour surface of a specified
threshold value of the vorticity modulus,

$\left|\bm{\omega}(\bm{x},t)\right|=\sqrt{\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}}$
(8)

in terms of its Cartesian components. Vorticity-based criteria are
particularly appropriate as the essential characteristic of any turbulent flow
is a time-dependent stochastically varying vorticity field, which is logically
a more fundamental parameter than the scalar concentration field of a flow-
visualizing dye, often used in experiments (e.g. Westerweel  _et al._ (2005),
Westerweel  _et al._ (2009), Dimotakis  _et al._ (1983)). The definition of
the edge is determined as a characteristic value of $|\bm{\omega}|$ connected
with the interface layer of relatively rapid change that is encountered as a
test point moves inward toward the jet core from the ambient (nearly)
irrotational flow (see Fig. 1). Based on extensive studies of these vorticity
profiles across the edge (Shinde et al (2018)), it is found convenient to
introduce two distinct edges or boundaries of the jet. The first is located at
the ‘middle’ of the T/NT interface at $|\bm{\omega}|=0.5$, beyond which
$|\bm{\omega}|$ varies rapidly to 1.0 or above in the turbulent core. This
defines the ‘inner’ boundary of the turbulent jet.

For reasons that will be clear shortly, we find it necessary to define an
‘outer’ boundary at $|\bm{\omega}|=0.1$, which similarly locates the
rotational / irrotational (R/IR) boundary of the jet. The basis for this
choice will not be presented here in detail, chiefly because the images shown
in the rest of the paper provide ample supporting evidence for the choices
made (see Sec. IV). The contour surfaces $|\bm{\omega}|=0.1$ and $0.5$ will
thus be referred to in the sequel as the outer and inner boundaries of the jet
respectively. There have been other proposals for defining two boundaries,
using slightly different criteria. Reeuwijk & Holzner (2014) van Reeuwijk and
Holzner (2014) define an inner and outer boundary that separate the turbulent
core from the buffer region and the VSL from the buffer region respectively.
The inner boundary in their analysis represents the enstrophy threshold for
which the interface propagation velocity $v_{n}$ is zero and the outer
boundary respresents the threshold below which the enstrophy production is
negligible.

In most of the images of jet cross-sections presented here, it has been found
illuminating to include a background vorticity field within the core of the
jet. Depending on the purpose of the analysis being carried out, different
choices may be made for the particular measure of vorticity that is most
appropriate for the analysis. Thus $\omega_{z}$ may be more relevant in a
diametral section. Similarly, in an axial section in the $x-z$ plane at $y=0$,
either the modulus $|\omega_{y}|$ or the signed value of $\omega_{y}$ or the
azimuthal component $\omega_{\phi}$ may be more appropriate.

The axial section is taken over the range $23\leqslant z\leqslant 45$. The
flow is also studied in two diametral planes, one at $z=34.05$ which lies
exactly in the middle of the self-preserving zone, and the other at $z=42.69$.

##  IV Jet Boundaries

We shall now present the results derived largely from an analysis of axial and
diametral sections. Particular attention is given to possible connections
between the flow-field outside the jet boundary and the vorticity field within
the jet. Only a limited number of 2D sectional data are analyzed, as this is
adequate for the conclusions drawn here. A more detailed analysis, including
that of selected 3D velocity and vorticity fields will be reported separately.

\begin{overpic}[width=455.24408pt,height=355.65944pt,unit=1mm]{Fig_box.eps}
\put(15.0,0.0){\includegraphics[width=369.88582pt]{Tst=185000_vel_vort_mag_Phi=0_th_0p1.eps}}
\end{overpic} Figure 2: Axial section in the $xz$ plane at $t=925$, showing
the instantaneous velocity and total vorticity modulus
$|\bm{\omega}(\bm{x},t)|$ fields. The black and pink velocity vectors
(respectively with radially inward and outward components) are shown only in
the ambient beyond the outer boundary of the jet, represented by the red
contour at $|\bm{\omega}|=0.1$; the orange contour at a threshold of 0.5
represents the inner boundary). For comparison two typical flow velocity
vectors (blue) inside the core of the jet near the jet centerline are plotted.
To avoid clutter, only every fourth velocity vector in the $x$-direction and
every third in the $z$-direction are plotted. The red vertical arrow at the
bottom-left corner is a reference velocity vector of magnitude $0.1w_{0}$.
\begin{overpic}[width=469.47046pt,height=182.09746pt,unit=1mm]{Fig_box.eps}
\put(0.0,0.0){\includegraphics[width=116.65646pt]{Jet_L_int_Tst=185000_Phi=0.eps}}
\put(41.0,0.0){\includegraphics[width=106.69783pt]{Jet_R_int_Tst=185000_Phi=0.eps}}
\put(80.0,0.0){\includegraphics[width=239.00298pt]{Tst=185000_vort_mag_prof_phi_0_Z_ind=465.eps}}
\put(1.0,61.0){(a)} \put(81.0,61.0){(b)} \end{overpic} Figure 3: (a) Axial
section in the $xz$ plane at $t=925$. The $|\bm{\omega}|=0.1$ and $0.5$
contours in the axial cross section of Fig. 2 showing the wide variability in
the separation distance between the two contours that bound the yellow region;
the lighter reddish region is the turbulent core, the lighter lavender region
is the ambient. The red, blue and dark green boundaries are marked
respectively at thresholds $|\bm{\omega}|=0.1,0.5,1$. The thick red horizontal
lines at the bottom in the left panel show $5\eta$ (short line) and $5\lambda$
(long line). (b) A typical variation of the normalized instantaneous vorticity
modulus $|\bm{\omega}|$ (in red), compared with the time-averaged vorticity
modulus, $|\overline{\bm{\omega}}|$ (in blue). The bumps in $-10<x<-5$
correspond to the two intersections of the line $z=34.05$ with the outer
viscous buffer zone. The broken horizontal lines are marked at 0.1 (green) and
0.5 (pink).

The axial sections of the jet flow used in the analysis below are within the
self-similar region of the jet ($27\lesssim z\lesssim 40$). We begin with
Figure 2, which shows an axial cross section of the jet with contours of
$|\bm{\omega}|$ in the turbulent core. The yellow circles circumscribing the
numbers indicate the sub-regions (henceforth referred to as ‘SR’ followed by
the number) where significant entrainment activity appears to be taking place.
Following the inner boundary, we can discern in Fig. 2 a complete coherent
structure with a base along a 2-6 line (i.e., joining SR2 and SR6), narrowing
to a small nose around $z\approx 38$ penetrating the larger downstream
structure (of which we see only the bottom third) at its base along the line
4-9. These have a strong resemblance to structures observed in experiments
(e.g., Mungal & Hollingsworth Mungal and Hollingsworth (1989)) and other
numerical simulations (e.g., Basu & Narasimha Basu and Narasimha (1999)).
There is a similar penetration by the upper third of the upstream structure
along the base 2-6 of the central structure. The core of the central structure
$(30\lesssim z\lesssim 38)$ slopes slightly rightwards whereas the upper
structure has a tilted base 4-9 sloping up towards the left. On the whole, the
core of the jet appears to be in a helical mode (of the kind observed by
Dimotakis et al. (1983) Dimotakis  _et al._ (1983), Mungal and Hollingsworth
(1989) Mungal and Hollingsworth (1989)). It is known from studies like the
wavelet analysis of Narasimha et al. (2002) Narasimha  _et al._ (2002) that
the base of the structure is likely to be a fluted vortex ring with ambient
fluid below the base being drawn into the structure through the Biot-Savart
relation.

Figures 3 (a) and (b) respectively present axial sections near the edges of
the jet and the distribution of $|\bm{\omega}|$ across a diameter of the jet.
From Fig. 3 it is seen that there is a relatively rapid change in
$|\bm{\omega}|$ values as the T/NT interface at the inner boundary
$|\bm{\omega}|=0.5$ is crossed by a test point moving towards the turbulent
core. However, it is also seen that the separation between the contours of
$|\bm{\omega}|=0.5$ and $|\bm{\omega}|=0.1$ can vary appreciably at certain
locations along the boundary of the jet. For example, in Fig. 3(a) the inner
and outer boundaries are close to each other in some areas (e.g., $-8<x<-6$,
$38<z<45$) and well separated in others (e.g., $-8<x<-3$, $31<z<38$). In the
latter case we shall call the region between the inner and outer boundaries an
‘outer buffer zone’, to be distinguished from the buffer zone defined by
Reeuwijk and Holzner van Reeuwijk and Holzner (2014) in a plane jet, which
possibly is inwards of the present inner boundary, and will here be called the
‘inner buffer zone’. The threshold values used for defining the boundaries by
Reeuwijk & Holzner (2014) van Reeuwijk and Holzner (2014) are not applicable
here as these thresholds are a function of Reynolds number and also of the
dimensionality and nature of the flow. Figure 3(a) shows how the distance
between the two boundaries varies along the inner boundary of the jet. A
measure of this distance is the separation along the $x$ axis from the inner
to the outer boundary on either side of the jet. It is seen that this inter-
boundary separation varies from $0.090$ to $3.67$ in the left boundary and
from $0.077$ to $2.13$ in the right boundary.

For comparison, estimates of the two relevant scales in the problem, namely
the Kolmogorov length and the Taylor microscale, are shown in Fig. 3(a) at
$z\approx 33$, $x,y=0$. It is seen that inter-boundary separation varies from
the order of the Kolmogorov scale to the Taylor microscale; however, the
changes in $|\bm{\omega}|$ from 0.5 toward 1.0 and above in the turbulent core
is almost always sharp, and has a length scale comparable to the Kolmogorov
scale. Figure 3(b) makes this point by displaying the variation of the total
vorticity modulus $|\bm{\omega}|$ at $z=34.05$. For comparison the magnitude
of the mean flow vorticity is also shown in the figure. It is first of all
seen that the fluctuating vorticity $|\omega|$ in the core can be an order of
magnitude larger than the mean. Furthermore, while the T/NT interface at
$x=6.5$ drops steeply from about 4.0 to nearly 0, the interface near $x=-5$
drops steeply all the way to about 0.5, then becomes less steep, and finally
takes the form of two noticeable but smooth bumps, the one farther away being
the bigger. From comparison with Fig. 2 and Fig. 3 (a) the bumps correlate
with the outer buffer zone with its small but apparently non-turbulent
vorticity (as we shall shortly demonstrate). In the light of the many
discussions on this issue in the literature da Silva and Taveira (2010); da
Silva and dos Reis (2011); da Silva and Pereira (2008); Westerweel  _et al._
(2009); Hunt  _et al._ (2006) we conclude that both scales seem to have a role
to play, from the sharp T/NT interface layer of thickness $\mathcal{O}(\eta)$
to the larger lateral extent of the viscous buffer zone with a dimension of
order $\lambda$. Understanding the role of these different scales and whether
this difference in scales persists at higher Reynolds requires further
investigation.

Nature of the vorticity field from the T/NT to the R/IR interface

Beyond the outer boundary at $|\bm{\omega}|=0.1$, the flow is nearly
irrotational. It is seen (Fig. 2) that the core of the jet has $|\bm{\omega}|$
values ranging from 1.0 to more than 17 over much of the cross-section, except
in the very short section of length $\Delta x\simeq 2$ at $z\simeq 25$ to 28.
This could mark the region of separation between two successive coherent
structures in the jet, as may be inferred from jet flow visualizations by the
PLIF technique Dimotakis  _et al._ (1983), or by examining the computed
vorticity field of Basu & Narasimha Basu and Narasimha (1999) in a temporal
round jet. Another such separation between two successive structures is seen
in Fig. 2 around $z\simeq 38$ to 40. Here, the base of the structure is on the
4-9 axis, and clearly carries a vortex ring, as we can infer from the velocity
field in the ambient. Taken together, this lends further support to the view
that the flow exhibits a helical mode.

As expected, the edges of the jet are highly convoluted (especially the inner
boundary, which is a fractal curve (Sreenivasan & Meneveau (1986) Sreenivasan
and Meneveau (1986) and Sreenivasan et al. 1989 Sreenivasan  _et al._ (1989)).
Figure 2 includes the velocity field in the $xz$ plane outside the inner edge
of the jet. The arrows in dark gray indicate a radially inward component, and
those in pink a radially outward component. A few long arrows within the core
(with large heads) are also shown, to give an idea of the relative values of
the velocity magnitudes within and outside the boundary of the jet. It is seen
that in most of the selected sub-regions the velocity vectors crowd into a
part of the jet boundary (dark gray arrows), and represent entraining motions.
In some cases (above SR1, near SR4, between SR5 and 6) there are some pink
arrows as well, which could be detraining motions, but these are generally
locally diverging radially outward motions compared to the radially inward
converging and entraining motions. Whether the motions entrain or detrain also
depends upon the propagation velocity of the interface. However, Wolf et al.
Wolf  _et al._ (2012) and Reeuwijk & Holzner van Reeuwijk and Holzner (2014)
have shown that the interface velocity is of the order of the Kolmogorov
velocity, so generally negligible (at $z\approx 33$, Kolmogorov velocity is
$\approx 0.01\overline{w}_{0}$). It will be seen that there is little evidence
for a parallel inward flow in the ambient impinging on the jet boundary;
instead, entrainment often seems to occur in relatively intense events
concentrated in a few narrower areas near the edge (for example, below SR1,
SR4, SR7 and SR9; this is further discussed in section V). In these areas the
flow velocities can go up to $0.4$ times the mean centre-line velocity
$\overline{w}_{c}$ (e.g., in SR7). Thus the fluid is pushing the turbulent
edge inward and rushing into the turbulent core across the T/NT interface
through ‘gulfs’ or ‘wells’ created by the inrush of ambient fluid. (This is
best seen in movies of the flow; see the short sequence of stills in Fig. 4.)
In all these cases the streamlines in the outer flow exhibit, in part, one or
two roughly circular patterns near the entraining areas, suggesting that they
represent a velocity field induced by relatively strong vorticity in the near
core (see analysis of SR9 in sec. V). Occasionally, however (see e.g. the
region between SR2 and SR3), the flow appears to be radially organized,
indicating a source-like flow, which we shall further discuss below.

\begin{overpic}[width=455.24408pt,height=247.53888pt,unit=1mm]{Fig_box.eps}
\put(5.0,44.0){\includegraphics[width=199.16928pt]{Tst=186000_vort_mag_phi=0_th_0p1.eps}}
\put(76.0,44.0){\includegraphics[width=199.16928pt]{Tst=188000_vort_mag_phi=0_th_0p1.eps}}
\put(5.0,0.0){\includegraphics[width=199.16928pt]{Tst=190000_vort_mag_phi=0_th_0p1.eps}}
\put(76.0,0.0){\includegraphics[width=199.16928pt]{Tst=193000_vort_mag_phi=0_th_0p1.eps}}
\par\put(148.0,17.0){\includegraphics[width=28.45274pt]{Tst=186000_vort_mag_phi=0_th_0p1_colorbar.eps}}
\put(12.5,51.5){{(a)}} \put(83.5,51.5){{(b)}} \put(12.5,7.5){{(c)}}
\put(83.5,7.5){{(d)}} \end{overpic} Figure 4: Temporal evolution of a part of
the jet from Fig. 2. Three areas in the figure labeled as A, B and C
illustrate important features of the evolution (see text for discussion).

Figure 4 shows the time evolution of a part of the jet $(32<z<36)$ that is
accurately self-preserving. Region A hardly changes in shape or in vorticity
range over the whole time interval $930\leqslant t\leqslant 965$ covered by
the images. On the other hand, region B keeps continuously shrinking, and has
disappeared at $t=965$. This disappearance is largely due to the outward
motion of the inner boundary till, in (d), the two boundaries are very close
to each other in the interface above $z=34$. Region C undergoes minor changes
in shape, in part again because of the outward movement of the inner boundary.
Between (a) and (d) it is seen that the turbulent core has expanded, with a
general reduction in the larger vorticity-sparse areas seen near the
boundaries in the earlier images.

\begin{overpic}[width=227.62204pt,height=147.95424pt,unit=1mm]{Fig_box.eps}
\put(5.0,0.0){\includegraphics[width=184.9429pt]{Time_Trace_vort_mag_phi=0_tog.eps}}
\end{overpic} Figure 5: Temporal variation of the total vorticity modulus at
two points in the plane $z=34$, one on the centerline $(0,0,34)$ and the other
at $(-8,0,34)$ in the outer buffer region between the inner and outer
boundaries.

Figure 5 compares the variation of $|\bm{\omega}|$ with time at two points.
The vorticity at the center line shows substantial variation in time, whereas
that in area A in the outer buffer zone is more than an order of magnitude
lower and exhibits a slow and gentle decay with hardly any fluctuation that
can be associated with turbulence. This justifies why such a buffer zone can
be characterized as ‘viscous’, and the structure at zone A in Fig. 4 may be
called a ‘viscous tongue’, most probably a relic or fossil from an earlier
excursion of in- or out-of-plane vorticity from the core or buffer zone into
an ambient nearly at rest.

\begin{overpic}[width=455.24408pt,height=307.28984pt,unit=1mm]{Fig_box.eps}
\put(0.0,49.0){\includegraphics[width=122.9161pt]{Tst=185000_vel_vort_azim_Z=42p69_th_0p1_half.eps}}
\put(45.0,49.0){\includegraphics[width=183.52019pt]{vel_vort_azim_Tst=185000_Phi=0_th_0p1_SR_9_SupZoom.eps}}
\put(111.0,49.0){\includegraphics[width=142.54814pt]{Tst=185000_streamlines_phi_0_SR9_Z=42p5.eps}}
\put(19.0,0.0){\includegraphics[width=170.71652pt]{Tst=185000_inv_vel_VERT_phi_0_fit.eps}}
\put(81.0,0.0){\includegraphics[width=170.71652pt]{Tst=185000_inv_vel_HORI_phi_0_fit.eps}}
\put(0.0,103.0){(a)} \put(44.0,103.0){(b)} \put(108.5,103.0){(c)}
\put(14.0,43.0){(d)} \end{overpic} Figure 6: (a & b) Instantaneous velocity in
the ambient fluid and azimuthal vorticity $(\omega_{\phi})$ field at $t=925$.
Red and blue colors indicate clockwise and counterclockwise vorticity
respectively. The red and green lines show the outer and inner jet boundaries
respectively at thresholds 0.1 and 0.5 based on $|\bm{\omega}|$. The green dot
is nominal center of the red vorticity patch (see main text). Two ticks on
either side of zero on the color-bar correspond to $+0.1$ and $-0.1$. Black
vectors show inward velocity, while the pink vectors indicate outward flow
velocities. The reference velocity vector shown in red color at the bottom-
left corner of (a) is $0.05w_{0}$ and at the bottom-right corner of (b) is
$0.025w_{0}$. To avoid clutter, only every seventh velocity vector is plotted
in (a) and every third in the $x$-direction in (b). (a) A portion of the
diametral section at $z=42.69$ showing a vortical object at the right around
$x=8$. (b) Zoomed-in view for SR9. (c) Instantaneous streamlines for SR9
starting from $z=42.5$. The pink curve marks the streamline separating the
fluid spiraling into the vortex from that which drifts away from the vortex.
The dashed grey rectangle marks the zoomed portion shown in (b). (d) The
inverse velocity relation with distance along the horizontal (left panel) and
vertical (right panel) lines drawn in (b) showing a region of $1/r$ decay like
that due to a line vortex normal to the $xz$ plane at the location of the
green dot (nominal centre).

##  V Structure of the axial sections at selected subregions

To analyze the relation between the vorticity field in the turbulent core and
the velocity field beyond the R/IR boundary, we consider selected subregions
in axial and diametral planes. We start with the axial sections, namely SR 9,
1, 4 and 7.

SR9: This offers the simplest flow situation among those seen in Fig. 2 for
the following reasons illustrated in Fig. 6. (Note that the back-ground
vorticity field displayed in Figs. 6 \- 8 is the signed azimuthal component
$\omega_{\phi}$, which is helpful in correlating the velocity field in the
ambient with the vorticity field in the core.) The dominant feature of the
flow within the jet core in SR9 in Fig. 2 is a large lump of clockwise
(negative, blue) vorticity. A diametral section of the jet around SR 9 at
$z=42.69$ is shown in Fig. 6(a). By comparing with Fig. 2, it is seen that
this diametral section of the core flow in SR 9 contains an ovoid around
$x\approx 8,~{}-1<y\leq 1.5$, with $x,y,z$ dimensions of about $2\times
5\times 2$. Thus the variation of the vorticity with $y$ in the central region
of the ovoid is not very strong. From Fig. 6(b), which is a zoom on SR 9 in
Fig. 2, it is seen that the highest azimuthal vorticity
($|\omega_{\phi}|=6.8$) is confined to a nearly circular region of diameter
less than 1.0. The maximum in the other components of vorticity are about a
fifth of that of $\omega_{\phi}$. Although the structure is 3D, the azimuthal
component dominates over the other components of vorticity, so to a first
order approximation we can treat it like a 2D flow. The image clearly
indicates that there is a circulating motion around the vortex in a direction
that is consistent with the sign of the vortex. Very close to the region where
vorticity is intense and of one sign over the region (as often happens in
coherent structures) there appears to be relatively high correlation between
the vorticity and velocity fields. Figure 6(b) thus depicts a strong
circulatory motion, first hugging the outer boundary and then penetrating the
inner boundary.

Figure 6(c) shows instantaneous effectively $2D$ streamlines in the $xz$ plane
for SR9 starting from points located along $z=42.5$, corresponding to the flow
field in Fig. 6(b). Note that all these streamlines start from neighboring
grid points (closer to each other in the core than in the ambient, see Sec.
II), and therefore do not represent equal differences in the stream function.
The red and black contours are the same as in Fig. 6(b). A ‘dividing’
streamline, separating fluid which spirals back into the vortex from fluid
that moves away, is shown in pink. Such an interpretation is encouraged by the
weak variation of vorticity in the $y$-direction, making the flow close to a
2D field as already noted.

From Fig. 6(c), with vorticity in the object peaking at $x=8.28$, $y=0$,
$z=42.69$, the domination of the azimuthal vorticity in determining the
velocity field in the neighborhood is understandable. This is further
supported by Fig. 6(d), which shows a plot of the reciprocal of the magnitude
of the $w$ and $u$ components of the velocity respectively in the $xz$ and
$yz$ planes, as a function of radial distance $r$ from an approximate origin
along the lines $z=42.69$ and $x=8.28$ respectively (shown by cyan colored
lines in Fig. 6b). A linear fit is a good match to the data over the range of
$r$ between 1.6 and 3.2, as marked by dashed red lines; the equations for the
fits are shown on the respective plots. This supports the conclusion that the
outer velocity field approximates to that induced by a line vortex along the
$y$-axis, by the Biot-Savart law. By extending the fitted line to the zero of
the inverse of the velocity, we can identify an “effective center” for the
vortex (marked by pink squares in Fig. 6d). The fact that these effective
centers are not identical but not very far from each other (separation about
0.3) is consistent with the finite size and the imperfect symmetry of the
vortical structure seen in Fig. 6(a).

SR1 & 2: Figure 7(b) (the azimuthal vorticity field alone is shown in Fig.
7(a)) illustrates how the ambient velocity field maybe understood in terms of
two anticlockwise vortices next to each other near the inner boundary. In SR2
the vorticity is anticlockwise near the edge but clockwise away from it
further inward. The net effect of this somewhat more complex vorticity field
is that in the region $28<z<29$, $-7<x<-6$, the flow in the ambient is away
from the jet (pink arrows), with only a mild contribution to the inward
circulatory motion. In the zone $26.5<z<27.5$, however, the motion is more
strongly circular in the lower parts of SR2.

\begin{overpic}[width=469.47046pt,height=398.33858pt,unit=1mm]{Fig_box.eps}
\put(0.0,68.0){\includegraphics[width=244.69386pt]{vort_azim_Tst=185000_Phi=0_th_0p1_SR_1_2.eps}}
\put(84.0,68.0){\includegraphics[width=221.93158pt]{vel_vort_azim_Tst=185000_Phi=0_th_0p1_SR_1_2.eps}}
\put(0.0,0.0){\includegraphics[width=221.93158pt]{vel_vort_azim_Tst=185000_Phi=0_th_0p1_SR_4.eps}}
\put(80.0,0.0){\includegraphics[width=239.00298pt]{vel_vort_azim_Tst=185000_Phi=0_th_0p1_SR_7.eps}}
\put(0.0,136.0){(a)} \put(80.0,136.0){(b)} \put(0.0,63.0){(c)}
\put(80.0,63.0){(d)} \end{overpic} Figure 7: Ambient velocity and core
azimuthal vorticity $(\omega_{\phi})$ fields, showing the inrush of flow into
the core of the jet from the ambient fluid. All other conventions are the same
as in Fig. 6(b). (a) SR1, SR2 $(\omega_{\phi})$ field (b) SR1, SR2, with
inrush velocity field in the ambient with the $(\omega_{\phi})$ field; the
tick marks shown above and below zero on the color-bar indicate respectively
$\omega_{\phi}=+0.1$ and $-0.1$ (c) SR4, (d) SR7. Magnitude of the reference
vector shown in red at the bottom-left corner of (a, c, d) is $0.025w_{0}$.
Color-bar shown in (b) also applies to the vorticity contours shown in (a, b,
c, d).

SR4: Here (Fig. 7(c)) the vortices in the upper reaches $38<z<42$ resemble the
region near SR2 in Fig. 7(a). Because of the weaker clockwise component in the
vertical strip $38.2<z<41.8$, $-7.5<x<-6.5$, the flow in $37<z<40$ is
dominated by counterclockwise vorticity.

SR7: Fig. 7(d) is the most complex image among the ones selected here for
analysis. There is first of all the clockwise motion at the lower levels of
SR7 ($31.5<z<32.2$, $3.5<x<6.5$). Similarly there is a counter-clockwise
circulation in the upper levels of SR7 ($34.3<z<36$, $3.5<x<6.6$). It is clear
that the former is strongly influenced by the large region of clockwise
vorticity in the lower half of SR7, and the latter is attributable to the
clockwise circulation due to the vorticity of the same sign in the block
$34<z<36$, $4.5<x<6$. The large values (up to $0.4\overline{w}_{c}$) of the
inrush velocities here may compensate for the small area at the entrance to
the gulf ($\sim d_{o}$), and hence can make a non-negligible contribution to
the entrainment rate.

The above four examples show some of the different types of organized motion
that occur due to interactions between vortex elements that are part of the
coherent structures in the core fluid, and the Biot-Savart induced velocity
due to these elements at the T/NT interface. The most striking feature of the
flow in the ambient is that it is organized largely in terms of interacting
circulatory motions. Roughly speaking the diameter of the region containing
this ordered outer motion is about 20 $d_{0}$ at $z\simeq 40$ to 44, and about
12$d_{0}$ at $z\sim 28$. This is approximately twice the local diameter of the
turbulent core. These estimates of the spatial extent of the velocity and
vorticity structures involved in the interaction are a strong indication that
they are related to elements of the large-scale vorticity in coherent
structures in the core flow, and cannot be attributed to small scale
vorticity.

This analysis may be compared with two others in the literature. Phillip and
Marusic (2012) Philip and Marusic (2012) show, in a ‘caricature’ of a
turbulent jet and the entrainment (their Fig. 6), a flow of the ambient fluid
towards the turbulent core in parallel streamlines almost till they impinge on
the edge of the jet, in contrast to the ordered motion we see here in Figure
2. Westerweel et al. (2009) Westerweel  _et al._ (2009) showed an
instantaneous velocity field (Fig. 19 in their paper) that appears (in one
part of the domain covered in the diagram) to transport turbulent fluid out of
the turbulent core across the interface into the ambient. However, in a frame
of reference fixed with respect to the vortex within the core, the velocity
vectors were almost parallel to the interface. Thus, they concluded that
organized motions do not transfer irrotational fluid into the turbulent core.
In the immediate neighborhood of the vortex, however, the relative velocity
induced in the ambient fluid appears to be crossing into the core. It is thus
seen that the picture presented by the present DNS analysis is quite different
from conclusions drawn by experimental results which may be affected by the
limitations of PIV measurements in the low-velocity irrotational region.

##  VI Analysis of diametral sections

Figure 8 presents a sequence of images of the diametral section at $z=42.69$
at four different instants ($t=925,950,960$ and $970$). As before the ambient
velocity field is presented against a background of the vorticity field
$\omega_{\phi}$ within the turbulent core.

\begin{overpic}[width=469.47046pt,height=495.07777pt,unit=1mm]{Fig_box.eps} \put(0.0,94.0){\includegraphics[width=231.88979pt]{Tst=185000_vel_vort_azim_Z=42p69_th_0p1.eps}} \put(82.5,94.0){\includegraphics[width=231.88979pt]{Tst=190000_vel_vort_azim_Z=42p69_th_0p1.eps}} \put(0.0,13.0){\includegraphics[width=231.88979pt]{Tst=192000_vel_vort_azim_Z=42p69_th_0p1.eps}} \put(82.5,13.0){\includegraphics[width=231.88979pt]{Tst=194000_vel_vort_azim_Z=42p69_th_0p1.eps}} \put(26.0,0.0){\includegraphics[width=52.63777pt]{Tst=185000_vel_vort_azim_Z=42p69_ref_vect.eps}} \put(74.0,0.0){\includegraphics[width=231.88979pt]{Tst=185000_vel_vort_azim_Z=42p69_colorbar.eps}} \put(10.0,6.0){\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>\put(10.0,2.0){\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>\put(156.0,5.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(0.0,170.0){(a)} \put(83.0,170.0){(b)} \put(0.0,89.0){(c)} \put(83.0,89.0){(d)} \end{overpic} Figure 8: Four instantaneous diametral sections of images at $z=42.69$ at times $t=$ (a) 925, (b) 950, (c) 960, (d) 970. The vorticity field in these images is the signed azimuthal component $\omega_{\phi}$. Thick black vector indicates velocity inside the jet at $~{}(0,0)$. The two ticks, one on each side of zero on the color-bar, correspond to $+0.1$ and $-0.1$. To avoid clutter only every seventh velocity vector is plotted. \begin{overpic}[width=469.47046pt,height=495.07777pt,unit=1mm]{Fig_box.eps} \put(0.0,94.0){\includegraphics[width=231.88979pt]{Tst=185000_vel_vort_axial_Z=42p69_th_0p1.eps}} \put(82.5,94.0){\includegraphics[width=231.88979pt]{Tst=190000_vel_vort_axial_Z=42p69_th_0p1.eps}} \put(0.0,13.0){\includegraphics[width=231.88979pt]{Tst=192000_vel_vort_axial_Z=42p69_th_0p1.eps}} \put(82.5,13.0){\includegraphics[width=231.88979pt]{Tst=194000_vel_vort_axial_Z=42p69_th_0p1.eps}} \put(26.0,0.0){\includegraphics[width=52.63777pt]{Tst=185000_vel_vort_azim_Z=42p69_ref_vect.eps}} \put(74.0,0.0){\includegraphics[width=231.88979pt]{Tst=185000_vel_vort_axial_Z=42p69_colorbar.eps}} \put(10.0,6.0){\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>\put(10.0,2.0){\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>\put(156.0,5.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\put(0.0,170.0){(a)} \put(83.0,170.0){(b)} \put(0.0,89.0){(c)} \put(83.0,89.0){(d)} \end{overpic} Figure 9: Four instantaneous diametral sections at same $z$ and $t$ as in Fig. 8. The background vorticity field is the signed axial component $\omega_{z}$. See Fig. 8 for additional details. Zone |  Flow events based on images in Fig. 8(a-d) and an image at $\mathbf{t=975}$ not included in Fig. 8
---|---
A |  $\bullet$ Mild inrush at $t_{0}=925$, gets deeper but bigger, begins to connect with H by $t=950$
B |  $\bullet$ Starts with outflow at $t_{0}$.
$\bullet$ Boundary pushed outward by vorticity in the turbulent core,
gradually disappears by $t=975$
C |  $\bullet$ Starts at $t_{0}$ with mild inrush, grows slowly and covers CD region in a broad inrush by $t=970-975$
D |  $\bullet$ Radial outflow prominent at $t_{0}$, decays slowly, disappears at $t\simeq 970/975$.
$\bullet$ Boundary is advancing throughout.
E |  $\bullet$ Modest outflow capped by inrush at bottom of D and lower part of E at $t_{0}$.
$\bullet$ Upper part from D weakens, outflow disappears by $t=950$.
$\bullet$ Lower E squeezed by expansion of core.
$\bullet$ Weak outflow reappears at E at $t=975$.
F |  $\bullet$ Mild inrush at $t_{0}$, little change even at $t=975$.
G |  $\bullet$ Modest outflow at $t=925$, pushed out by both inner and outer boundaries (which remain close to each other).
$\bullet$ Spans almost whole quadrant [from $(-2,-6)$ to $(7,0)$] by $t=955$.
$\bullet$ Shrinks by $t=965$, making room for new inrush at H, but still
active at $t=975$.
H |  $\bullet$ Inrush moves right, gets weaker by $t=955$.
$\bullet$ A and H inrushes appear to coalesce (from opposite directions), and
become a major feature of the flow by
$t=970/975$, like C.
Table 1: Flow events across a diametral plane

The inner and outer boundaries of the jet are indicated as before by contours
of $|\bm{\omega}|$ at 0.5 and 0.1 respectively. As with axial sections, we see
in Fig. 8 regions where the inner and outer boundaries are very close to each
other, as well as other regions where they are much farther apart. The flow
field in the ambient is of two distinct types, the dominant one being radially
outward motions, with circulatory motion also present but not as strong as in
the axial section of Fig. 2. The radial motions often issue from those regions
where the inner and outer boundaries are close, for the simple reason that
core-flow vorticity elements are then close to the ambient fluid. For the same
reason circulatory motions may also be present in such an area (see for
example Fig. 6, and also the area straddling the $x$ axis at $y=0$ across the
first and fourth quadrant in Fig. 8).

The diametral section shown in Fig. 8(a) is just above the wide base S4-S9 of
an existing coherent structure, the lower part of which is seen near the top
around $z\simeq 38$ to 42 in Figure 2. The strong source-like radially outward
motion seen in the ambient fluid across the second and third quadrants,
$-9<x<-5$ which we call zone A, has a striking degree of circular symmetry,
and which is not entirely consistent with the azimuthal component of vorticity
shown in Figure 8(a). A plausible explanation involves the effect of
streamwise vortcity which has not been considered so far. Figure 9(a) shows
data at the same stations as in Fig. 8(a), except that at each of the four
stations, the azimuthal component of vorticity is replaced by the streamwise
component. Figure 9(a) shows the presence of a pair of streamwise vortices in
zone A (red:counter-clockwise and blue:clockwise). The arrangement of these
streamwise vortices along with the local azimuthal vorticity result in an
outward movement of the fluid between the vortices from the turbulent core
towards the T/NT interface, which in turn results in an instantaneous
displacement effect (note that the inner and outer boundaries are very close
to each other and are pushed outward along with the fluid). This boundary
expansion is also evident from a comparison between Fig. 8(a) and Fig. 8(b).
Interestingly, a somewhat similar radial flow is noticeable in the fourth
quadrant of Figs. 8(a-d). Also, the vortical object of SR9 (Fig. 6) is clearly
seen in the diametral section in Fig. 8(a) where its length along the $y$ axis
is seen to be about 5. However, the feature gradually changes its shape with
time, and is substantially different at $t=970$ (Fig. 8(d)). Between the data
in Figs. 8(a) and Fig. 6(a), it can be confirmed that the vorticity in the
object around the point $x\approx 0,~{}y\approx-7$ is confined to a $2\times
5\times 2$ ovoid.

It may be noted that the in-rush events, so conspicuous in the axial section
shown in Fig. 2 and Fig. 6, are substantially weaker in the diametral section.
This suggests that the major contribution to entrainment is made by the inward
and upward velocity induced in the ambient fluid due to elements of azimuthal
vorticity in the coherent structure in the core.

To understand the evolution of the flow events in the near ambient of the
turbulent core, the flow region in the diametral plane shown in Fig. 8 is
divided into 8 approximately equal semi-quadrants A to H in the $xy$ plane,
e.g. starting with A in the angular sector $0^{\circ}<\phi<45^{\circ}$ around
the origin $(x,y)=(0,0)$ and going round anticlockwise to H in the sector
$335^{\circ}<\phi<360^{\circ}$. The flow shows some very interesting features
as can be clearly seen in Figs. 8(a) to (d), and documented in Table 1. The
turbulent core is largest at $t=950$ (Fig. 8(b)), consistent with the fact
that the widest part of the coherent structure, with its base SR4-SR9
containing a vortex ring, is passing through the plane $z=42.69$ at the time.

##  VII Relation between vorticity field at T/NT interface and its near-
neighbourhood

In most of the previous work, the boundary between the turbulent core and the
ambient fluid has been determined by thresholds on either scalar
concentrations Westerweel  _et al._ (2009); Burridge  _et al._ (2017);
Westerweel  _et al._ (2005) or an out-of-plane vorticity component Chauhan
_et al._ (2014). Boundaries defined by the former have often been the subject
of controversy mainly because of the vast difference between the diffusivity
of the scalar and viscosity. Furthermore, the evolution equation of the
passive scalar and vorticity are fundamentally different due to the absence of
the stretching-tilting term in the former. One could argue that the gradient
of a passive scalar and vorticity are orthogonal to each other in the limit of
very high Peclet and Reynolds number as was shown in a recent work by
Patwardhan and Ramesh (2018) Patwardhan and Ramesh (2018). They show that
scalar gradient surface tracks the vorticity field if the two were aligned
initially. However, they also showed that in the regions with sharp gradients
the gradient of passive scalar and vorticity are not orthogonal. We know that
at the T/NT interface the velocity and scalar gradients are sharp and as a
consequence a scalar cannot faithfully represent the turbulent flow field. So
vorticity is fundamentally the more logical criterion to use, but to determine
all three components of the vorticity vector experimentally is difficult. The
3D results from DNS studies can therefore provide valuable insights from data
on the full vector $\bm{\omega}$.

We have shown here that the use of $|\bm{\omega}|$ sheds much light on the
subject. The contour $|\bm{\omega}|=0.5$ is close to the turbulent core of the
jet, and is a good candidate for the threshold defining the inner edge of an
interface and is consistent with the choice $\omega_{y}\simeq 0.5$ made by
Westerweel et al. (2009) Westerweel  _et al._ (2009) for the out-of-plane
vorticity component ($\Omega_{z}$ in their notation). Thereafter
$|\bm{\omega}|$ drops steeply going outward; the $|\bm{\omega}|=0.5$ contour
is approximately at the point where the outward gradient of $|\bm{\omega}|$ is
steepest in the T/NT interface(i.e., Corrsin’s viscous superlayer), and is
taken here as defining the inner boundary of the jet.

The nature of the irrotational flow in the ambient near the T/NT interface has
not received enough attention. With the present simulation more precise
pictures of the flow in the ambient have been constructed. These pictures show
a considerable degree of organization or order in the flow, usually
circulatory in nature but occasionally radial as well. It is often (but not
always) possible to trace the circulatory motions as those associated with the
presence of elements of organized large-scale vorticity in the turbulent core
near the edge of the jet, making a highly plausible connection through the
Biot-Savart relation. Given this connection, it is not surprising that the
inner boundary of the jet, close to the region of transition from one
structure to the next in the core flow, often shows sharp and intense
incursions at velocities of order $0.4\overline{w}_{c}$ as already pointed
out. The highest velocity in the outer field is $0.34\,\overline{w}_{c}$ at
$x\approx 0.3$ and $z\approx 29.7$ in SR6 of Fig. 2. Axial sections of the
flow show that along either inner boundary of the jet, about 4 or 5 ‘inrush
events’ of the above type and others (at an average streamwise spacing of
$~{}5$) can be inferred. Although the length scale of a hypothetical ‘lid’ for
such an incursion into the inner boundary covers only a small fraction of the
surface area of the jet, the higher inrush velocities compensate to some
extent for the lower area, and could contribute more significantly to the
total entrainment than has been estimated in recent work (less than 8%
according to Westerweel et al. (2009) Westerweel  _et al._ (2009)). The DNS
images leave no doubt however that a significant part of the entrainment
occurs through inrush events which have an episodic character, reminiscent of
the momentum flux events that contribute to the Reynolds shear stress in a
high Reynolds number atmospheric boundary layer (Narasimha et al. 2007
Narasimha  _et al._ (2007)).

In summary, these observations suggest the following physical picture.

  1. 1.

The propagation - outward or inward - of the inner boundary, as well as the
organized motion of the ambient fluid that rushes inward, can result in
penetration that pushes inwards ‘vulnerable’ parts of the outer boundary. This
process is encouraged by the Biot-Savart induced velocity in the ambient fluid
due to elements of the vorticity field associated with the coherent structures
in the core, leading to what has been called engulfment.

  2. 2.

As long as there is a sharp boundary between turbulent and non-turbulent flow,
entrainment into the core flow must involve fluid crossing a fractal inner
T/NT boundary (which may itself be propagating into non-turbulent fluid); this
could legitimately be called ‘nibbling’.

It would be interesting to find out how the radial and axial components of
velocity at the T/NT interface, which are primarily responsible for the fluid
entrainment from the lateral and cross-sectional sides of a turbulent jet, are
correlated to the relavant velocity components at the surfaces of a short
circular cylinder circumscribing the T/NT interface. There is entrainment also
in the axial or streamwise direction, e.g. just below S4-S9.

We first choose the radial components for analysis. Consider the diametral
section at $z=34.05$, in the middle of the self-preserving zone. We now choose
a cylinder of height $6d_{0}$ ($32\leq z\leq 38$), and a radius of $8d_{0}$
which is just adequate to lie close to but outside the T/NT interface almost
everywhere around the circumference. Radial velocity vectors are plotted at
both the T/NT interface and the cylinder circumference at the selected instant
$t=2050$ as shown in Fig. 10; vectors in magenta and green indicate
respectively radially outward (generally detraining) and radially inward
(generally entraining) velocities (neglecting the interface propagation
velocity, see section IV). The contours within the T/NT interface show axial
velocity.

The procedure for finding the correlations mentioned above (along with its
limitations) is now outlined. The angular circumference of the disc (360
degrees, also periphery of the T/NT interface) is divided into 72 sectors,
each subtending $5^{\circ}$ at the center as shown in Fig. 10; this gives us
enough sectors to make reliable estimates of the correlations.

\begin{overpic}[width=455.24408pt,height=193.47882pt,unit=1mm]{Fig_box.eps}
\put(40.0,0.0){\includegraphics[width=227.62204pt]{Xind=465_Tst=410000_overlap_Uvel_th_0p5.eps}}
\end{overpic} Figure 10: Conventions followed for overlap analysis; see text
for details. Diametral section at $z=34.05$ and $t=2050$ showing the contours
of axial velocity and the velocity vectors at the T/NT interface and the
cylinder circumference. The reference velocity vector shown in red at the
bottom-left corner is $0.05w_{0}$. Velocity vectors at the disc circumference
are scaled up by a factor of four relative to the vectors at the T/NT
interface. Note that radially outward motions are rare and radially inward
motions dominate.

We use the following conventions while calculating the correlation:

  1. 1.

Correlated: If velocity vectors at both the T/NT interface and the cylinder
are in the same direction, either inward or outward, then that event is said
to be “correlated”.

  2. 2.

Anti-correlated: If velocity vectors at the T/NT interface and the cylinder
are in the opposite directions, either inward at the T/NT interface and
outward at cylinder or vice versa, then that event is said to be “anti-
correlated”.

  3. 3.

Non-correlated / non-decisive: We classify the event as “non-correlated /non-
decisive”
(i) if the velocities are negligibly small ($<0.05w_{c}$). By inspection and
trial, the threshold was determined to be about $0.001w_{c}$; for example,
look at the sectors between $185^{\circ}$ to $195^{\circ}$ in Figure 10;
(ii) when it is difficult to decide about the correlation because the velocity
at either T/NT interface or the outer circle is zero or very small ($\sim
0.001w_{c}$); for example, the velocity vectors at cylinder circumference in
sector between $330^{\circ}$ to $335^{\circ}$ in Figure 11(a);
(iii) when the T/NT interface goes beyond the outer circle, which has been
found to occur for three time instants viz. $t=2100,2125,2150$ out of 8 time
instants for less than about $40^{\circ}$ in the total of $2880^{\circ}$
studied, i.e., 1.4% of the circumference.
(iv) When the T/NT interface is highly convoluted, correlations are estimated
using the velocity vectors at T/NT interface closest to the cylinder, and
ignoring the inner T/NT interface. Figure 11(a) presents one such convoluted
T/NT interface.

\begin{overpic}[width=455.24408pt,height=187.78836pt,unit=1mm]{Fig_box.eps}
\put(0.0,0.0){\includegraphics[width=227.62204pt]{Xind=465_Tst=415000_overlap_Uvel_th_0p5.eps}}
\put(81.0,0.0){\includegraphics[width=227.62204pt]{Xind=465_Tst=435000_overlap_Uvel_th_0p5.eps}}
\put(8.5,60.0){(a)} \put(89.5,60.0){(b)} \end{overpic} Figure 11: See Fig. 10
for conventions. (a) Instantaneous image at $t=2075$ indicating the difficulty
in the selection of highly convoluted T/NT interface for finding the
correlations, especially in the fourth quadrant. (b) An instantaneous event at
$t=2175$ showing strong entraining correlations between the radial velocity
vectors at T/NT interface and cylinder circumference.

The eight time instants chosen for analysis are separated from their
neighbours by 25 flow units adding upto a total duration of 200 flow units. We
find that the velocity vectors are correlated for 73% of the circumference,
anti-correlated for 24%, and non-correlated for 3%.

Figure 11(b) shows one instant when the velocity vectors at T/NT interface and
cylinder circumference both inward and strongly correlated. Note that 65 out
of 72 sectors (more than 90% of the circumference) are correlated and inward,
while 7 out of 72 sectors are anti-correlated (i.e., less than 10% of the
circumference). This suggests that entraining motions at the T/NT interface
are strongly correlated with the radially inward component of the velocity at
the cylinder’s surface.

##  VIII Mass Flux Budget

This section presents the net mass flux budget within the turbulent core
determined by $|\omega|=0.5$ and also within the cylindrical disc at
$t=1995.25$ to demonstrate the accuracy of the mass flux entrainment
calculations. This is done by calculating the streamwise variation of the mass
flux in the turbulent core and correlating it with the mass flux entrained
into a fixed cylindrical disc of radius $8d_{0}$ and height $\approx 6.5d_{0}$
circumscribing the T/NT interface. A schematic of the disc and the T/NT
interface is shown in Fig. 12. The volume between the cylindrical disc surface
and the T/NT interface will be called the sheath. The mass flux at the
cylinder edge is computed as

$\displaystyle\int\textbf{u}\cdot d\textbf{S}$ (9)

where $d\textbf{S}$ represents the local surface area vector at the edge of
the cylinder. The mass flux in the turbulent core is computed using Eq. 9
where $d\textbf{S}$ now represents the local area vector in the streamwise
direction. The coordinates of the interface were detected with the iso-surface
function in MATLAB using the criterion $|\omega|=0.5$. The detected interface
was mapped onto the nearest grid points in the computational domain, thus
pixelising the interface. The advantage of this mapping is that the mass flux
budget is accomplished to a very high accuracy as shown below and is not
affected by any interpolation errors. The results are as follows:
Mass flux budget: cylindrical disc
Mass flux into the disc from the bottom surface ($m_{z_{1}=32.35}$)
$=~{}7.472$
Mass outflux from the top surface of the disc ($m_{z_{2}=38.64}$) $=~{}7.959$
Mass flux into the disc from the lateral boundary ($m_{NT_{o}}$) $=~{}0.486$
Net mass flux into the disc accounting for surface
normal=$m_{z_{1}}-m_{z_{2}}+m_{NT_{o}}$ $\approx~{}10^{-11}$

\begin{overpic}[width=455.24408pt,height=301.59938pt,unit=1mm]{Fig_box.eps}
\put(30.0,0.0){\includegraphics[width=284.52756pt]{schmt_mass_budget.eps}}
\end{overpic} Figure 12: Schematic for the mass flux budget (see text for
details). Color code: red for turbulent motion (T), green for non-turbulent
motion (NT).

Mass flux budget: turbulent core
Mass flux into the turbulent core from the bottom surface
($m_{T}(z_{1}=32.47)$) $=~{}7.327$
Mass outflux from the top surface of the turbulent core ($m_{T}(z_{2}=38.51)$)
$=~{}7.126$
Mass outflux from the top sheath surface ($m_{NT}(z_{2})$) $=~{}0.798$
Mass flux into the bottom sheath surface ($m_{NT}(z_{1})$) $=~{}0.155$
Mass flux into the lateral boundary of the cylinder ($m_{NT_{o}}$) $=~{}0.443$
Total mass flux from the sides of the cylinder
($m_{NT_{o}^{*}}$)=$m_{T}(z_{1})-m_{T}(z_{2})+m_{NT}(z_{1})-m_{NT}(z_{2})=~{}0.443$
Mass flux balance=$m_{NT_{o}}$ \- $m_{NT_{o}^{*}}$=$10^{-7}$

It must be noted that the $z$ location of the mass flux budget in the core and
the cylinder is slightly different due to the staggered location of velocity
and vorticity. This results in a slightly different mass flux from the lateral
boundary of the cylinder. From the numbers given above its is seen that the
entrainment mass flux calculations at the T/NT interface are accurate up to
$10^{-7}$ and the continuity equation up to $10^{-11}$. At $t=1995.25$, for
$32.5\leq z\leq 38.5$, we note that the disc had a net mass flux entraining
into the cylinder and the turbulent core had a net mass flux detraining from
the core. This would seem to suggest that the dynamics at the T/NT interface
and the cylinder are not in sync as opposed to the conclusion derived in the
previous section. This may well be due to the fact that conclusions in the
previous sections did not account for the magnitude of these events. However,
it must be noted that the present inference is based on the analysis of a
single time instant. A more detailed analysis is required to understand the
entrainment dynamics close to the T/NT interface. We also compute the Taylor
entrainment coefficient $\alpha_{E}=\frac{1}{2\pi
W_{c}b_{we}}\frac{d\overline{m}}{dz}$ I. (1945) using the data averaged over
1750 flow units where $\overline{m}$ represents the mean mass flux within the
turbulent core, $W_{c}$ is the mean centreline velocity and $b_{we}$ is the
mean velocity width, defined by $w(b_{we})=W_{c}/e$. The entrainment
coefficient in the self-preserving region in the present data is 0.065 which
is smaller than the $\alpha_{E}=0.074$ at $z\approx 34$ reported in Carazzo et
al. (2006) Carazzo  _et al._ (2006). This was expected, as the present
calculation does not account for the mass flux outside the turbulent core, in
contrast to the analysis in Carazzo et al. (2006). Additional details are
discussed in Shinde et al. (2019).

##  IX Entrainment as an episodic process

To study the nature of entrainment in a turbulent jet it is important to
understand how the irrotational ambient fluid enters into the turbulent jet.
Figure 13 shows the streamwise variation of the radial mass flux entering into
the lateral surface of the $8d_{o}$ disc at various instants. We see that
entrainment varies in both space and time, and it can be spatially anti-
correlated in $z$. Thus, the net entrainment in the region $32<z<35$ at a
given instant is anti-correlated with the net entrainment in $35<z<39$. Figure
14 shows the variation of streamwise mass flux in the turbulent core, obtained
by integrating the streamwise velocity over the area within the T/NT interface
at any $z$. The anti-correlation observed at the edge of the cylinder is not
as strongly visible in the streamwise mass flux data. Some of the mass flux
entrained into the cylinder may be retained in the non-turbulent region
outside T/NT interface. We note that the mass flux in the turbulent core of
the jet increases between $34\leq z\leq 38.5$ over the time span shown in Fig.
14. This is not surprising as it is linked to the passage of large scale
coherent structures. The presence of these structures locally enhances
entrainment events that explains the increase in the mass flux in the
turbulent core. And in the absence of these structures the mass flux in the
core decreases. The decrease in the mass flux in the core is visible in the
region between $32.5\leq z\leq 34$ for the duration $2030\leq t\leq 2045$.
This trend would propagate downstream at later times, and the cycle repeats.
Similar observations were reported in the recent studies on turbulent plumes
Burridge  _et al._ (2017); Plourde  _et al._ (2008). To understand the nature
of these observations it is necessary to examine the entrainment dynamics
close to the T/NT interface.

Figure 13: Streamwise variation of radial mass flux into the $8d_{0}$ diameter
cylindrical surface. Time indicated in flow units. Figure 14: Streamwise
variation of the mass flux within the T/NT interface ($|\omega|=$0.5). Time
indicated in flow units.

Figure 15 shows the azimuthal variation of mass flux at the R/IR interface at
intervals of $2^{\circ}$. In certain angular regions there exist several
interface co-ordinates due to the contortions in the interface. At these
locations only the mass flux spatially averaged over $2^{\circ}$ is plotted.
Figure 15 shows that there are instances when localized regions of very high
mass flux entraining into or detraining from the turbulent core. At such
times, the mass flux contributions from the positive events are significantly
larger than from the negative events. From Fig. 15 and earlier images (e.g.,
see Fig. 2) it appears that entrainment is episodic. We identify about 13 to
16 such events at each $z$ location shown in Fig. 15. Some of these events or
episodes may be spatially correlated. To better describe this episodic
character we use the algorithm described by Narasimha et al. (2007) Narasimha
_et al._ (2007) where they adopted a threshold method to identify the
productive and counter-productive periods in an atmospheric boundary layer
momentum flux time series. Following their analysis we adopt the r.m.s. value
$m^{*}$ as an appropriate unit for the threshold, and an event is detected if
the instantaneous mass flux $m(t,z,\phi)$

$\displaystyle|m|>km^{*},$ (10)

where $k$ is a constant to be selected. This is done by examination of the
data at the T/NT interface at intervals of 2 degrees in the azimuthal angle
($\phi$). The angular extent of an event in the azimuthal direction is defined
as a continuous interval in the azimuthal angle where Eq. 10 holds true. This
entire interval is then considered as one spatial event at a given instant.

Figure 15: Azimuthal variation of mass flux at the R/IR interface (threshold
level on $|\omega|=$0.1). The green lines indicate $\pm$ 30% of r.m.s of the
mass flux. Figure 16: Variation of fractional contribution to mass flux at the
R/IR interface and the corresponding cumulative duration as a function of
threshold $k$ for event detection based on Eq. 10. Figure 17: Variation of
cumulative mass flux as a function of cumulative duration for various event
detection thresholds $k$. The events are arranged in descending order of the
(signed) magnitude for calculating cumulative mass flux and cumulative
duration.

We define a few additional parameters to characterize the events. The
amplitude of an event $i$ is defined as

$A_{i}(t)=\frac{1}{\overline{m}}\int_{\phi_{i\hat{1}}}^{\phi_{i\hat{2}}}\frac{m(t,z,\phi)}{\phi_{i\hat{2}}-\phi_{i\hat{1}}}d\phi,$
(11)

where $\overline{m}$ is the mean mass flux (averaged over time and space)
across the T/NT interface and $\phi_{i\hat{2}}-\phi_{i\hat{1}}$ is the angular
extent of event $i$. Thus the amplitude is the ratio of the contribution from
an event to the mean value of the total net mass flux. The fractional mass
flux contribution from an event $i$ is given by

$m_{i}=A_{i}*\frac{\phi_{i\hat{2}}-\phi_{i\hat{1}}}{T},$ (12)

where $T$ is the total length of the data set in degrees, over an appropriate
number of realizations. So the net contribution of an event is a product of
both amplitude and duration. Every event has a finite extent over time and
space. Note that the events are not tracked over time, only the azimuthal
extent of the event at a given time is accounted for in the present analysis.

To determine the flux events the mass flux data at the R/IR interface were
analyzed in $34<z<35$ for a total time of $185$ flow units. In Fig. 16, the
ratio of the cumulative contribution of detected events to the total mass flux
for various values of the threshold is presented. The objective of this plot
is to determine the appropriate threshold required to detect the minimum
number of events that will account for almost the whole flux to describe the
mass flux balance. From Fig. 16 such a threshold is around $0.3$, as it
accounts for about $99.5\%$ of the total mass flux.

Narasimha et al. (2007) Narasimha  _et al._ (2007) also used a burstiness
parameter to determine the compactness and significance of the detected
events. The contribution from each event is calculated using eq. 12. The
events are arranged in descending order of magnitude of their contributions
from the largest positive through to the largest negative contribution. The
cumulative contribution from this re-ordered sequence of events is then
plotted against the cumulative duration of the events as shown in Fig. 17. The
burstiness curve reaches a maximum when the contributions of all the
entraining (positive) events have been integrated, and then decreases thereon
due to the detraining (negative) events. The curve is plotted for various
thresholds from 0.0 to 1.0. For a threshold greater than about $0.3$, the
detected events do not account for the total mass flux. This would suggest
that for higher thresholds the weaker events are required for a complete
description of the interfacial dynamics. For thresholds less than $0.3$,
contributions from sub-threshold events are like noise with zero mean with
little net contribution to the net mean mass flux $\overline{m}$. For the
threshold of $0.3$, the total contribution from positive events reaches about
$1.77\overline{m}$ over 37$\%$ of the total duration, and that from the
negative events is about $0.77\overline{m}$ over about $22\%$ of the total
duration. Following the terminology from Narasimha et al the positive events
are called productive and the negative events counter-productive. The rest of
the time, over which the net contribution is negligible, is called the idle
phase.

Figure 18: Burstiness curve: Variation of cumulative mass flux as a function
of cumulative duration for an event detection threshold $k$ of $0.3$. The
events were ordered in the descending order of the absolute magnitude, for
calculating the cumulative mass flux and the cumulative duration.

We may also choose to arrange the events ordered according to the absolute
magnitude. With such an arrangement, the burstiness curve never exceeds unity
and has small discontinuities, as shown in Fig. 18. Following Narasimha et al.
(2007), the burstiness factor $B$ is then defined as the ratio of the area
between the curve and the line AO to the area of the triangle ABO. If $B=0$
for an event series, then the burstiness curve is represented by the dashed
line AO. This indicates that all events have equal contributions over equal
durations for the entire dataset. On the other hand, if $B=1$, then the
burstiness curve would be the vertical axis BO, which implies that the events
are like a series of delta functions with zero net duration, i.e. all the
contributions occur instantaneously with zero duration with the amplitude
approaching infinity such that the net contribution is finite. For the present
data shown in Fig. 17, the burstiness factor is $0.75$, which indicates that
there are significant intermittent periods of intense activity at the R/IR
interface that dictate the net mass flux entraining into the turbulent core.

The visual and quantitative anaylsis presented here so far suggests that the
velocity field outside the T/NT interface is organized and are induced by the
large scale ordered motions in the turbulent flow. The irrotational fluid is
drawn towards the T/NT interface by the large scale coherent motions and not
by viscosity-dominated nibbling. Moreover, the induced flow field could either
be entraining or detraining depending on the organization and orientation of
the vortices near the T/NT interface. The induced flow near the T/NT interface
contorts the interface, and the irrotational fluid appears to be entraining
into the turbulent core through spatially and temporally localized intense
events wherein viscous action imparts vorticity to the irrotational fluid as
it crosses the T/NT interface. Thus entrainment is a two step process. In the
first step, which is also the rate limiting step, irrotational fluid is driven
by the large-scale dynamics into the turbulent core through the gulf or well-
like regions in the T/NT interface, and in the second step vorticity is
imparted to fluid parcels through viscous processes as they cross the T/NT
interface.

The two-step entrainment process discussed above is consistent with the views
expressed by Plourde et al. (2008), Philip & Marusic (2012), Mistry et al.
(2016) and Burridge et al. (2017). Mistry et al. conclude that entrainment is
a continuous multiscale process wherein at large scales the entrainment
velocity is high to account for the smooth surface area and at smaller scales
the entrainment velocity is relatively low, transporting flux occurring across
a highly contorted fractal surface. Burridge et al. analyse the data on plumes
and highlight the influence of the large scale motions in the entrainment
process through conditional statistics. They show that vertical mass transport
is decreased or increased in the absence or presence of large scale eddies and
conclude that these eddies impart significant vertical velocities to the
ambient fluid near the T/NT interface. We relate it to motions induced by
vortices in the coherent structures, especially the toroidal vortex at the
base of the structure. Moreover, Burridge et al. also show that the horizontal
velocities are significantly enhanced in the absence of large scale eddies
which in our analysis correspond to the region between the toroidal vortices
of successive/neighbouring arrow head shaped coherent structures as in Fig. 2.
The azimuthal variation of mass flux near the T/NT interface discussed in the
present paper points to the episodic nature of entrainment. Further study will
be required to determine whether all events may be associated with the large
scale motions close to the T/NT interface and also to describe the shape of
these events. It may be worthwhile to track the motions of the passive
particles which originate from the irrotational region and study how they
acquire vorticity near the T/NT interface.

##  X Concluding remarks

The objective of the present paper has been to describe the entrainment
process in a round turbulent jet, using data provided by a well-resolved DNS
study. As a preliminary step towards achieving this objective, detailed
analyses of the DNS data have been made in the near-neighbourhood of the
turbulent/non-turbulent (T/NT) interface, on the ambient as well as in the
turbulent core neighbourhood of the interface. It is found necessary to
distinguish a rotational / irrotational (R/IR) interface from the T/NT
interface, the region in-between being rotational but not turbulent.
Interestingly, it is found that the region between the core and the ambient
have a length scale varying from the Kolmogorov to the Taylor length scales,
indicating that both scales are relevant. The flow in the ambient
neighbourhood of the T/NT interface is found to consist of several well-
ordered circulating motions. By the analysis of such a flow field in a
relatively simple case, wherein an approximately two-dimensional blob of
vorticity is present, it has been shown that the ordered motion in the ambient
flow is due to the induced velocity of vorticity elements in coherent
structures in the core-side neighbourhood. When and where the ordered motions
are relatively intense in the ambient, the T/NT interface is dented towards
the core, which may develop into a deep and twisted well or gulf. It is shown
that the entrainment process can be broadly divided into an engulfment phase
near the edge of the jet, followed by nibbling across the T/NT interface
deeper in the well, supporting a similar conclusion of Burridge et al. (2017)
regarding the episodic nature of the entrainment process. It is convenient to
think of this process as constituting an entrainment ‘event’, intermittent in
time and concentrated in space. These events are amenable to the method of
analysis used for momentum flux events in the atmospheric boundary layer
(Narasimha et al. (2007)). It is found that the burstiness of the signal is of
the order of about 75%, with entrainment about 40% of the time, detrainment
over about 10% of the time and very little activity over the rest of the time.
This picture suggests that for applications where greater entrainment is
desired, for example, one may have to devise means by which the intensity and
duration of entrainment events can be enhanced.

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