Journal of Applied Mechanics Vol.11, pp.869-879
Unsteady
Simulation
Md.
* PhD
Student
***Associate
using
numerical
of
is
shown
with
agreement
higher
process
normal
Key
non-linear
of
DNS
turbulent
Axial
two
vortex,
the
axial
data.
rotational
energy.
types
Trailing
velocities.
In
the
matured
of anisotropic
vortex,
as
well
vortex
non-linear
rate
observed
of
a trailing
RANS,
on
and
not
model
of axial
the
non-linear
the
vortex,
radial
self
examined
comparison
captured
and
by
causes
shows
velocity
in
time
The
are
Sapporo)
vortex
different
discussed.
growth
are
vortex,
the
were
decay
depending
(060-8623,
trailing
k-ƒÃ model,
the
stage
Kyoto)
profiles
turbulence
the
as
are
zones
behavior
Rankine
of
strong
that
time
a
Kyoto)
University,
circulation
standard
effects
and
(615-8540,
simulation
and
However,
different
of
(615-8540,
of a turbulent
decay
using
It is observed
Five
University
field
present
rapid
Means
Kimura***
Hokkaido
vortex
velocity
performed
extremely
velocity.
kinetic
show
in
by
University
velocity
of
tangential
Since
and
previous
tangential
stresses
Words:
model.
3D
features
is also
it predicted
tangential
with
than
of
simulation
Kyoto
Kyoto
for
captured
of
Ichiro
of Engineering,
out
general
distribution
The
k-ƒÃ model,
decay
is carried
The
**,
Management,
School
Vortex
Model
Management,
Graduate
growth/decay
radial
vortex.
standard
rapid
k-ƒÃ model.
in the
trailing
Hosoda
*, Takashi
of Urban
and
turbulence
similarity
Ali
of Urban
JSCE
Axial
k-ƒÃ
simulation
non-linear
periods
for
, Faculty
Turbulent
Non-Linear
Shahjahan
, Department
Professor
Unsteady
of
, Department
** Professor
(August 2008)
a
a
good
is much
growth/decay
the
turbulent
distance.
k-ƒÃ model.
1. Introduction
zero at the center as well as far away from it. In addition to
these basic vortices, there are other time dependent rotary
The basic types of plane vortices can be classified into
two categories: one is slower velocity at center and
motions that have azimuthal velocity component as well as
radial and axial components.
maximum at sides; and another is maximum at center and
minimum velocity at edges. The rotary fluid motion of first
one is called the solid body rotation, since it is similar to the
fluid motion filled in a rotating hollow box. On the other
hand, if a long circular rod rotates in a fluid with constant
The existence of such vortices is not limited to natural
activities but also in many engineering applications. The
Rankine vortex has been used extensively in various studies;
for instance, to predict the decay of wing-tip vortices, to
estimate the noise level produced by vortices and vibrations,
velocity around its axis, the fluid velocity can be found
highest and equal to the velocity of rod at the rods surface
to model the natural phenomena such as hurricanes and
(due to adhesion); and with increasing distance from the rod,
the velocity is observed to be diminished in inverse
partially governed by the vortex wake hazard caused by the
proportional to the distance. Such a fluid motion is called a
potential vortex. Fluid motion composed of a potential
vortex and solid body rotation is called Rankine vortex after
the fluid dynamicist Rankine. The radial distance from
center to the maximum tangential velocity is called the
radius of vortex core. For a steady circular motion without a
tornados. Current aircraft spacing in and around airports is
possible interaction of leader aircrafts trailing wake and
vortices with a following aircraft. However, the vortex
behavior behind a lifting vehicle is still a topic of debate.
The understanding about the internal structure as well as the
mechanism of turbulence growth in the vortex core is also
debatable. Vortex flows are the major cause of cavitation
velocity component normal to the plane of rotation, the
and underwater acoustics in marine propellers. Cavitation
and hollow core due to pressure deficit at vortex center
Rankine vortex in the only possible vortex whose velocity is
causes air-entraining vortices, which is an important part in
―869―
hydraulics.
only
Therefore,
due
to
importance
emphasis
paper
with
vorticity
wing
VĮ(tangential)
two
and
vortex
is the
presence
well
known
that
line
is
Navier
equation
tool
Because,
or
adopted
strain
of
non-linear
differs
from
i) the
account
of
added
tuned
in
the
of
normal
flow
the
inertial
effects
Hoffmann
to
believed
solid
of
that
of
to follow
body
a behavior
of
of
but
boundary
nearly
core.
This
velocity
high
this
strain
region
layer
log
wake
diffusion.
tangential
that
The
the
the
rotation
negligible
reported
turbulent
by
between
maximum
are
& Joubart7)
rotation.
turbulent
region
the
point
body
described
to include
buffer
and
the
be
near
rate
is very
wall,
and
law.
most
2.
Non-Linear k-ƒÃ
Model
2.1
of
the
field
its
basic
equations
flow
in
are
a k-ƒÃ
model
for
an
unsteady
as follows.
equation:
most
(1)
having
Momentum
hand,
in
this
in two
important
a constant
and
stress
stresses.
(2)
the
used
parameters;
equation:
a
(cƒÊ) is not
(3)
ii)
equation
The
realizability
in simple
The
Continuity
capturing
Reynolds
equations
incompressible
isotropic
other
by
Basic
of
k-ƒÃ model
model
the
turbulence
where
inward
a
solid
can
two
resolve
of
of
result
considering
of
one
the
rotation
is
outer
similar
averaged
flow
k-ƒÃ model
and
anisotropy
anisotropy
the
coefficient
strain
are
the
are
the
standard
To
the
is
non-linear
viscosity
terms
constants
the
eddy
function
non-linear
and
superior
The
extended
contains
exist.
memory
standard
On
at
the
to.
because
viscosity2).
but
nearly
(Region-III)
Region-II
region
improvement
in
for
potential
clarification
of
the
results
predicts
the
k-ƒÃ model
rotation
turbulence.
are
attention
However,
and
eddy
model
anisotropic
the
a
as
computer
appeared
satisfactory
of
assumption
a
term
one1).
produce
rate
paid
tip of
produces
region
applications1).
and
Therefore,
still
The
Averaged
engineering
areas
a
trailing
such
model,
time
and
equations,
frequently
viscosity
most
is
component.
(Reynolds
stress
limitation
be
the
models,
CPU
stress
Navier-Stokes
study
where
outer
circulation
velocity
RANS
DNS.
should
Reynolds
ways:
not
components,
between
of axial
practical
and
at the
constant
difference
the
less
the
models
but
for
LES
possibility,
high
is
practical
which
created
, and
Reynolds
used
to
cannot
its
velocity
turbulence
it requires
compared
the
topic
to
vortex,
non-zero
main
type
model
popular
to
the
Stokes)
trailing
Vz (axial)
Therefore,
RANS
due
concentration
with
and
the
infinity.
the
present
but
diffusion
deals
dependent
semi-infinite
It
on
interest,
as well.
This
time
the
academic
ƒÃ
- equation:
model
conditions
shear
flows
(Ali
et al. 3), 4)).
In
this
velocity
study,
field
non-linear k-ƒÃ
using
model.
are
previous
the
captured
data.
circulation
In
forming
profiles
both
experimental
the
examined
studies,
a rolling
extensively
and
are
in
results
DNS
up
for
trailing
the
turbulent
vortex
sheet
experimentally
and
numerical
study
decay
as
The
p:
self
values
used)
compare
described
the
the
non-linear k-ƒÃ
Rankine
The
innermost
region
The
tangential
velocity
model
vortex
is
as
called
is
zero
trailing
vortex
been
at
are
The
by
2.2
Constitutive
(a)
Standard k-ƒÃ
In
Phillips5)
considered
predictions.
vortex
the
density
k
of
: the
=
energy
1.0, ƒÐƒÃ=
in
fluid,
averaged
cƒÃ1, cƒÃ2:
and
direction,
v
:
turbulent
1.3,
rate,
model
cƒÃ1 =
the
xi
dissipation
the
ui:
average
P:
the
molecular
energy, ƒÃ:
vt:
constants
1.44
and
the
the
eddy
(standard
cƒÃ2=1.92
are
core
center
equations
studied
numerically.
the
model
standard k-ƒÃ
model,
the
Reynolds
stress
tensor
equations
derived
ui from
here
is
solved
by
linear
constitutive
viscosity
concept,
Phillips5)
a multi-layered
as
respectively
turbulent
of ƒÐk
Ui
2).
uj
to
the
viscosity,
averaged
and
coordinates,
velocities
pressure,
the
spatial
viscosity, ƒÐk, ƒÐƒÃ,
velocity
have
Qin6)
turbulent
kinematic
with
growth/decay
observation
by
the
the
vortex.
and
theoretical
xi:
and
of
discussed.
tangential
(4)
where,
performed
of vortex
of
3D
using
results
turbulence
are
distribution
previous
from
of
also
simulated
features
zones
vortex
performance
the
simulations
radial
is
the
general
time
present
in
and
with
axial
simulation
comparing
The
simulation
a turbulent
The
by
different
in
similarity
for
model,
assessed
DNS
as
numerical
out
standard k-ƒÃ
models
well
unsteady
is carried
structure.
Boussinesq
(Region-I).
of
the
account
core
•\ 870•\
eddy
the
anisotropy
effect.
which
does
not
take
into
two
(5)
(either
S
generalized
vt
is
of k and ƒÃ,
determined
and
from
the
is approximated
dimensional
consideration
Recently,
authors
functional
considering
Here,
or Ħ).
the
proposed
form
effect
functional
of
forms
have
for
these
both
the
as
follows:
are
proposed
more
coefficients
parameters3),4).
The
by
(6)
Here,cƒÊ
(b)
bears
a constant
Non-linear k-ƒÃ
value
the
stress
constitutive
(13)
0.09.
model
Including
Reynolds
of
(14)
non-linear
equation
anisotropy
introduced
equations
can
be
term
by
in
the
Yoshizawa8),
expressed
in
the
the
Table
following
1. Values
of
coefficients
for
cƒÊ and
cƒÀ
forms
(7)
Here,
vt
is
constant;
and
also
cƒÀ is
SƒÀij are
determined
the
as
coefficient
defined
as
Eq.
of
(6),
but
non-linear
cƒÊ is
not
quadratic
a
term;
follows:
Here,
the
and
(8)
Cns,
CnĦ,
model
algebraic
It is known that the non-linear terms in equation (7) are
can
equivalent to the following mathematical formulation9),10).
some
stress
be
of
the
are
standard
Where,
the strain
and rotation
tensors
are defined
neglected,
as
this
terms.
study
the
to
effect
Eq.
are
14
(10)
Comparing Eq. (9) with the non-linear terms of Eq. (7), the
relations between the coefficients can be derived as
i.e.
the
of
given
above
gives
and
when
S=Ħ=0,
of
and
in
or
simply
the
Pope9)
neglecting
strain
and
cƒÊ becomes
the
mdĦ are
cƒÊ used
Speziale10)
the
Neglecting
standard
strain
CdsĦ1, Mds
form
equation
Moreover,
0.09.
CdĦ1
Gatski
the
neglected
of
Cds1
functional
by
from
reduces
If
CdsĦ,
model
value
model
CdĦ,
The
obtained
effects
(9)
Cds,
constants.
rotation
equal
quadratic
to
term,
the
the
k-ƒÃ model.
and
cƒÀ =
rotation
CƒÀ0. The
parameters
values
on
of
cƒÀ is
cƒÀ0 used
in
below:
(15)
2.3 Tuning of model constants and evaluation of model
performance
(11)
From
of
this
comparison,
eddy
viscosity
parameters.
(Ħ)
of
it
(cƒÊ)
The
are
in
and
Gatski
Pope9),
is
strain
defined
is
also
a
inferred
function
parameter
Eq.
(12),
and
that
of
(S)
as
used
strain
and
in
the
coefficient
and
rotation
the
anisotropy of turbulence in plane shear layer, and that in
rotation
Eq. (14) are tuned to satisfy the realizability conditions for
parameter
previous
The values of model constants are given in Table 1.
The model constants in Eq. (13) are tuned considering the
2D shear flows.
Further, the model is tested with some basic turbulent
studies
Speziale10).
flows, such as turbulent jet with and without swirl (Ali et
al.3)). The analytical results are compared to previous
experiments, and the model constants are tuned to obtain the
(12)
Many
the
kinds
coefficient
parameter,
Cotton
Kimura
Hosoda14)
functions
Most
rotation
Ismail11),
&
model
cƒÊ.
and
&
of
of
have
them
parameter
Kato
&
consider
been
consider
is
only
neglected
Launder12)).
one
proposed
Craft
dominant
et.
for
strain
(such
as,
al13)
and
parameter
of
•\ 871•\
best fitted comparison. The swirl jet calculation explores the
model's applicability for a flow field with high strain and
rotation rate. The applicability of the model is examined for
large scale vortices considering the spatial distribution and
topological change of turbulent structures with singular
points in an idealized vortex street (Ali et al. 4)).
We also simulated the 3D flow field for compound
open channel flows, and the performance of model is
evaluated from the view point of mean flow and bed shear
momentum
equation,
approximated
with
respectively.
k
and ƒÃ
considering the predictability of coherent vortices in the
equation.
interface of main channel and flood plain (Ali et al. 15)).
explicit
procedure
DNS
towards
of the vortex. Since the axial velocity of this vortex is not a
of
function of axial distance but a function of radial distance,
2D numerical grid became applicable to simulate 3D flow
field. The details of the flow field is given below.
Tangential
is
the
the
100
size
in
each
domain
overcome
the
is
is
time
same
for
hydraulic
taken
each
as
is
3-D
mesh
at
The
computational
lateral
fully
the
time
solved
using
with
and
Qin's
variable
and
y)
field
of
centre
large
boundary
velocity
domain
(x
sufficiently
of
by
in
conditions
grid
two
interference
achieved
with
field
for
step.
dense
of
used
discretized
pressure
are
schemes
is
accuracy,
successively
numerical
with
boundary.
grids
of
2D
used
scheme
simulations
under
computation.
spacing
difference
are
The
each
numerical
performed
The q-vortex, for which the direct numerical simulation
results are available (Qin6)), is considered as initial condition
step.
at
Unsteady
central
second-order
solved
by
fluxes
upwind
equations
and
step
diffusive
advancement
of
basic
in
iterative
Initial Conditions
Time
forms
increment
3.1.
central
scheme
The
and
and
hybrid
equations.
Adam-Bashforth
are
QUICK
The
stress profiles, structure of secondary currents as well as
3. Simulation Details
convective
grid
coarser
consists
directions.
The
(10m•~10m)
in
the
to
vortex
decay
process.
Cartesian
velocity
results
Radial velocity,(17)
,(16)
using
Axial velocity,(18)
grid
are
the
finally
geometric
is
used
presented
for
in
computation,
cylindrical
and
the
coordinate
conversion.
(a)
Here, Vois the scaling velocity, related to the initial mean
tangential flow, defined as
(19)
Mp,
as
is
the
used
initial
DNS
Qin's
simulation.
in
As
discussed
q-vortex
can
find
mach
that
unstable
and
to
velocity
deficit
the
to
stability
is,
chosen
et
a
of
to
and
the
al.16),
value
of
of•@
addition
axial
temporal
thus,
to
values
the
velocity
Lessen
related
any
=0.0009
y =1.1209
(b)
by
be
for
number
swirl
,
or
the
the
subtraction
profile,
or
excess,
does
vortex.
The
unity
to
an
of
They
initially
a
constant
inversion
not
initial
make
q.
is
of
velocity
the
number
vortex
velocity
be
stability
swirl
the
of
change
a
the
number
vortex
q0
initially
unstable.
0.
The
time
Here,
r0
velocity
is
is
the
contains
field
(at
t
=
independent
their
The
turbulent
0
are
Flow
).
initial
as
and
and
are
discretized
a
vortex
of
in
staggered
the
and ƒÃ.
initial
(c)
flow
is
In
t =
tangential
behavior
k
10-6
the
found
this
study,
respectively.
Computational
equations
on
(VĮm)
values
10-5
at
where
value
overall
the
governing
flows
The
by•@
distance
peak
given
domain
based
radial
the
on
values
3.2.
method
non-dimensionalized
schemes
for
mean
with
grid
velocities
the
finite
system.
and
Fig.1
volume
For
the
•\ 872•\
Distribution of (a) Tangential velocity (b) Axial
velocity and (c) pressure at t = 3.72T
4.
Results
rate is greater
than that for a laminar
vortex.
The distribution of tangential and axial velocities as
well as corresponding pressure distribution in the vortex
field are shown in Fig.1, for simulation time t=3.72T. The
case of trailing vortex considered here contains the axial
velocity with a minimum magnitude at vortex center (at
center, V, =0 at t=0) that gradually increases in outward
direction and became constant at a far distance from center.
In the simulated results, the axial velocity in the center
region is found to increase with time to reduce the gradient
with far field. The tangential velocity also decays with time.
The details of temporal change in flow characteristics are
explained in next sections.
The pressure in a vortex is not uniformly constant with
Fig.
2
Radial
the radial distance. We know the distribution of pressure for
a solid body rotation, for instance in a rotating box, is
decay
distribution
with
time
of
tangential
velocity
(Non-linear k-ƒÃ
and
its
model)
parabola17). The minimum pressure exists at center and
increases towards the wall of the box. On the other hand, for
a potential vortex the pressure increases from wall to
outward distance in a concave manner. Therefore, the
combination of these two gives the pressure distribution in a
line vortex. The distribution of pressure in a trailing vortex
is found similar to the line vortex regardless the presence of
axial velocity. The figure shows that the pressure at vortex
center is minimum and gradually increasing in outward
direction and became constant at a far distance from center.
4.1.
Decay
Figs.
and
using
for
than
the
causes
The
captured
extremely
rapid
decay
is
toward
the
value
at some
A circulation
initial
about
t >3.0T).
initial
after
Some
appear
Uberoi19)
argued
research
to
be
added
strong
Qin's
the
DNS
feature
the
pointed
enough
that
also
claimed
to
overshoot
that
to
that
of turbulent
is possible
(Non-linear k-ƒÃ
4 Radial
distribution
circulation
for
model)
and
found
a free
(at
ry/r0
matured
shows
a
(at
brief
unphysical
the
circulation
overshoot
a
times
normalized
adjustment
vortices.
produce
it
the
overshoot
the
of
and
core
the
is well
itself
effects
model,
reached
after
simulation
circulation
out
line
vortex
vortex
adjusts
Saffman18)
velocity
growth
that
the
different
of
velocity.
center
is observed
vortex
that
axial
direction
the
axial
turbulence
outside
when
standard
rotational
and
distribution
those
the
standard k-ƒÃ
at vortex
radial
radius
is a general
Phillips5)
and
the
a
7 show
and
Radial
calculated
that
tangential
Since
zero
5, 6 and
3
velocity,
times
observed
by
and
conditions.
overshoot
of
overshoot
conditions
overshoot
different
of tangential
circulation
of tangential
Figs.
is
model.
not
increase
stream
It
decay
non-linear
a rapid
for
model.
faster
were
profiles
velocity
model.
shows
predicted
of
axial
the
Non-linear k-ƒÃ
vortex
>5).
Fig.
4 show
standard k-ƒÃ
model
to
vortex
2, 3 and
circulation
by
of
visible
if vortex
Fig.
time
However,
is
In the
unlikely.
does
point
not
instability.
of
maximum
tangential
decay
velocity
•\ 873•\
maximum
velocity,
while
is at center
change
with
initial
velocity
and
its decay
with
decay
occurs
at the
model)
velocity,
maximum
percentage
diffusion
of axial
(Non-linear k-ƒÃ
of
time.
of
axial
vortex.
velocity
It is
observed
for
axial
Fig.
and
that
velocity
the
8 compared
the
peak
the
tangential
decay
rate
of axial velocity is much higher than tangential velocity.
Since the vortex takes some time to adjust with initial
condition, it is observed that the decay rate is small at initial
times, approximatelyt< 1.25T. For both the velocities,the
high decay rate is observed in intermediatetime period of
1.25T> t <4.25T. After that the decay slows down and the
slope of trajectories showsto proceed towards minimum.
Fig. 8
Comparison between time decay of axial velocity
and tangential velocity (Non-linear k-s model)
Fig.
5
Radial
decay
distribution
with
time
of
tangential
(Standard k-ƒÃ
velocity
and
its
model)
Fig.
9
Self-similar
compared
Fig.
6
Radial
different
distribution
times
of
(Standard k-ƒÃ
normalized
circulation
with
time
distribution
(Standard k-ƒÃ
of axial
circumferential
velocity
model.
model)
10 Self-similarity
with Phillips'
7 Radial
Phillips'
of
for
Fig.
Fig.
profile
velocity
and
its decay
with
model)
•\ 874•\
model
in the
circulation
profile
compared
4.2.
Self-similarity
From
showed
4.3.
of profiles
dimensional
analysis,
that for fully-developed
Hoffmann
&
Joubart7)
flow
Instability vortices
The non-linear model shows the occurrence of
instability in a well grown vortex field (i.e. at t >3.0T for
this case) after the self-similar form of flow field is attained.
(20)
Here,
r1
is
velocity
the
radial
VƒÆm. ƒ¡
1 is
its
value
By
is the
at
in
distance
that
a single
characteristics.
of
for
a
curve,
following
VĮ/ VĮm,versus
developed
with
to
and ƒ¡
flow
of
the
the
outer
experiments,
fit
velocity field from the main flow field. Phillips' multilayered structure is shown in the figure. Due to tangential
velocity gradient between region I and II, as well as between
of
independent
equations
=2ƒÎrVƒÆ,
r1.
fully
and
vortices in the flow field at t= 3.1T and 7.01T. Note that the
instability vortex field is calculated by subtracting the mean
tangential
as ƒ¡
results
Comparing
proposed
maximum
defined
experimental
showed
collapse
of
circulation
a radial
plotting
Phillips5
distance
Fig. 11 shows the flow velocity vectors and the instability
the
self
r/r1,
II and III, instability vortices are found to form in those
data
regions. Although at lower time (t=3.1T) the vortices
between region I and II are not visible, they are clearly
flow
observed at t=7.01T. In these figures, the instability vortices
Phillips5)
similar
are observed at a radial distance of about 0.7r1 and 1.2r,
respectively (locations are shown as dotted circular lines).
profile.
Region-I:
(21)
(22)
Logarithomic
part of Region-II:
(23)
In
self
Fig.
9,
it
similarity
Here
the
by
show
is
perfectly
self
in
form
only
fully
10
observed
that
region
II
the
self
studies
it
is
initial
and
self-similar
found
hydraulic
presence
support
of
the
self-similarity
region
reported
there
solution
near
Dissimilarity
in
does
previous
not
in
in
region
to
by
found
collapse
with
shows
conditions
the
and
in
for
III
and
(a) Calculated velocity vector of the flow field at t= 3.1T
in
is
region
component.
found
the
r/r1
=1
region
times
is
III
to
this
regardless
result
also
shows
t<3.0T.
as
in
circulation
for
on
that
II,
region
including
observed
scope
show
previous
noted
this
reported
Due
any
also
Present
that
show
dependent
I and
the
is
I and
not
From
weakly
It
region
does
(23)
Phillips
the
all
be
profile.
data.
I
studies.
seem
are
III
results
(22)
region
and
for
not
region
simulated
velocity
form
logarithmic
region.
axial
II
circulation
Eqs.
exists
argument
III.
results
I and
does
the
and
proposed
simulation
flow
conditions.
profile
II
(21)
region
curve,
the
that
III
simulated
form.
with
the
I,
Eq
region
Therefore,
a single
similarity
fitting
with
self-similar
the
in
3.0T.
velocity
regions
case.
the
although
t >
a particular
flow
collapse
excellent
the
for
shows
the
region
9.
tangential
all
t<3.0T,
but
Fig.
developed
Fig.
for
for
fitted
For
similarity,
shown
similarity
the
shown
model.
data
that
embraces
are
Phillips5)
the
observed
profiles
non-linear k-ƒÃ
to
is
solution
truly
The
universal
Figure
is
also
overshoot,
self-similarity
III.
(b) Instability vortices in the flow field at t= 3.1T
•\ 875•\
in its peak value and remains about constant throughout a
short period of zone (iv). It reveals that in this stabilization
period the flow field became saturated and cannot support
additional turbulence. Finally, the decay of turbulence is
started as predicted by most of the previous researches (such
as, Uberoi19), Bachelor20), etc.). The logarithmic plot shows
that the decay rate in zone (v) is much slower than the
growth rate in zone (ii).
(c) Calculated velocity vector of the flow field at t= 7.01T
Fig.
12
Growth
of
different
Fig.
normal
for
13
those
Here,
the
respectively.
, which
velocity.
in
times;
is
For
the
a
radial
axial,
are
Instability
vortices
in
the
flow
field
at
t=
Wygnanski
7.01T
a
Fig.
11
Non-linear k-ƒÃ
and
4.4.
axial
of
growth/decay
rates
radial
calculation6)
time
model.
kinetic
zones.
slowly,
as
initial
time
it
requires
in
decay
zone
the
(ii)
axial
and
time
period
velocity
(iii).
in
as
the
the
the
with
zone
(ii),
is
observed.
zone
gradient
The
of
vortex
the
relation
the
stresses
as•@
(r/r1<1)
and
on
up
to
•@
(Ali
et
matured
radial
show
distance.
distance
and
al.3),
stage
stresses
the
r1
tangential
jet
the
certain
is•@
by
surroundings,
of
in
no,
normalized
maximum
normal
depending
core
directions
stagnant
radius
turbulent
with
turbulent
tangential
of
a
the
of
is
However,
the
anisotropy
the
noted
in
=0.7
Inside
from
beyond
of
two
that
center,
distance
turbulent
any
reported
five
The
(iii).
It
kinetic
of
turbulence
In
nature
of
stresses
of
distribution
slows
that
high
in
reaches
•\ 876•\
et
the
al.22)
latter
and
case
is
to
of
turbulent
belongs
to
(t=
time
Comparison
of
it
increase
3.72
zones
turbulent
is
of
observed
significantly
central
the
in
&
Graham
zone
T,
5.0
(iii),
and
(iv)
intensities
that
all
In
are
7.013T),
and
(v),
between
only
of
by
23).
(ii),
stresses
T
an
studies
simultaneously.
normal
of
behaviors
previous
time
less
region
anisotropic
Phillips
stage
is
the
reported
found
times
in
These
also
growth
are
(iv),
as
wake.
different
and
production
such
are
Devenport
growth
is observed
energy
The
turbulence
diffusion,
axisymmetric
very
growth
significantly
where
than
of
different
changes
exponential
is
in
growth/decay
unphysical
an
situation
different
simulation
vortex
follows•@
and
five
energy
shows
(i)),
of
types
vectors
tangential
approximated
adjust
time
energy
next
of
to
In
kinetic
12
with
the
lifetime,
observed
(zone
velocity
field
to
kinetic
Fig.
energy
Initially
conditions.
turbulent
the
also
due
vortex
turbulent
is
non-linear k-ƒÃ
the
of
flow
and
distance
jet
Fielder21)).
vortex,
radial
distance
round
ry/r0
model)
time
is
In
of
the
with
turbulence
gradients.
turbulence
in
growth
velocity
DNS
prediction
vortices
Turbulence
Production
down
model
instability
&
trailing
at
distribution
radial
over
(d)
energy
(Non-linear k-ƒÃ
the
(Rii)
different
kinetic
periods
shows
stresses
=r0(t)
turbulent
time
normal
Fig.
13,
the
shown
for
which
are
respectively.
time
the
the
axial
zones
(iii)
velocity
component shows significant decay. On the other hand,
comparison between (iv) and (v), shows the simultaneous
decay of turbulent intensities in all the velocity components.
It is observed that at the center of vortex (i.e. at r =0),
throughout the simulation time. It confirms the
axisymmetric condition of the flow field.
Fig. 14 k-e model prediction of tangential velocity profile
compared with Qin's6) DNS results ( t = 3.72T)
Fig.
15 k-ƒÃ
with
Fig.
model
16 k-ƒÃ
velocity
Qin6)
at
time
effect
Also,
•\ 877•\
of
the
initial
this
of axial
DNS
velocity,
T
as
stage
flow
the
the
shown
in
vortex
and
shows
field
compared
profile
( t = 3.72T)
data
with
the
profile
velocity
results
Qin's
conditions
turbulent
= 3.72T)
compared
t =3.72
At
circulation
(t
DNS
with
are
respectively.
directions at (a) t 3.72 T (b) t 5.0 T (c) t-7.013T
Qin's6)
circumferential
axial
of
results
prediction
with
Comparison
The
Fig. 13 Radial distribution of turbulent intensities in three
DNS
model
compared
4.5.
prediction
Qin's6)
circulation
DNS
Figs.
14,
already
the
15
of
and
16
overcome
self-similar
became
and
calculation
the
behavior.
saturated
and
stabilized
after
gaining
passing
observed
that the
with
DNS
well,
the
velocity
high
shows
comparison
is
to the
profile.
in Fig.
faster
decay
for
17, it is observed
much
better
model
model
over
the
velocity
slightly
faster
Further
refinement
than
DNS
of the model,
constants
and/or
minimize
this small
addition
in
kinetic
decay slows down, and the slope of decay shows to proceed
towards minimum.
Five different
simulation.
kinetic
of
energy
model
model.
cause
A very
model
comparison
non-linear
the
and it is the
data.
DNS
the
standard
over-predicts
point,
DNS
and
in
For both the velocities, the high decay rate observed in
intermediate time period of 1.25T> t <4.25T. After that the
is fitted
tangential
standard
some time for initial adjustments, it is observed that the
decay rate is small at initial times, approximately t< 1.25T.
comparison
decay
the turbulence
that
comparison
slightly
maximum
than
the
observed
Comparing
well
of maximum
observed
much higher than tangential velocity. Since the vortex takes
energy
14 to 16, it is
velocity
position
also
kinetic
Figs.
shows
axial
non-linear
discrepancy
From
the
and
slightly
rate
of turbulent
model
Although
magnitude
circulation
value
periods.
non-linear
data.
decay
Such
a peak
its growth
shows
However,
energy
of decay
time
periods
are observed
in the
growth/decay profile of turbulence kinetic energy. Initially
the vortex changes very slowly to adjust with initial
conditions, then an exponential growth is observed that
the
slows down before gaining its peak value. The peak value of
its
turbulent kinetic energy remains about constant for a short
near
of tangential
data
as observed
in Fig.14.
period that finally follows the period of turbulence decay.
After the maturity of vortex, the turbulent normal
such
as fine tune of model
stresses show two types of anisotropic behavior depending
of 3rd order
terms,
are necessary
to
disagreement.
on the radial distance. Although the radial component is
always greater than other two, the tangential component is
greater near the center and smaller around the region of
maximum tangential velocity in comparison to axial
component. The decay rate of energy after the period of
stabilization is much slower than the growth rate in the time
zones before the stabilization. The decay of velocity field
slows down as the turbulence decay period starts.
References
1)
Fig.
17 k-ƒÃ
with
model
Qin's6)
prediction
DNS
of turbulent
results
kinetic
2)
energy
( t = 3.72T)
Jaw,
S.Y.
3)
closure
turbulence
ASCE.,
Vol.124,
Rodi,
W.,
in
by
Ali,
The
unsteady
isolated
non-linear k-ƒÃ
were
not
rapid
decay
non-linear
For
fully
of
fitted
overshoot
is observed
possible
It
with
the
k—s
W.
by
the
with
self
similarity
that
the
vortex
solution
the
curve
seems
roll-up,
time
6)
only
are
7)
to
8)
Circulation
field.
to
Due
be
not
in region III.
is
observed
decay
rate
of
axial
velocity
is
•\ 878•\
Qin,
vortex,
for
profiles
results.
grown
all
M.S.,
9)
Turbulent
I.
Flows
and
Onda,
axisymmetric
with
of Applied
Hosoda,
to
in
T.
predict
S.,
Swirling
jet
consideration
Mechanics,
scale
J.
Phd
Fluid.
The
of
JSCE,
I.,
Vol.
Mech.
105,
Purdue
of
vortex
pp.451-467,
University,
Applied
2007.
trailing
simulation
non-linear
of k-ƒÃ
turbulent
Journal
pp.723-732,
turbulent
A
change
vortices,
Numerical
thesis,
Kimura,
spatial
Vol.10,
W.R.C,
J.H.,
and
the
large
JSCE,
Phillips,
Engg.
2006.
Mechanics,
5)
J.
1979.
Kimura,
an
second
environmental
for
model
Journal
structures
data.
solution
of
non-linear k-ƒÃ
model
the
DNS
Ali,
for
pp.259,
T.,
solution
pp.821-832,
a
equations
circulation
simulated
well
for
model
and
causes
similarity
in a single
Phillips'
4)
it predicted
However,
Phillips)
9,
I: overview.
Methods
Kollmann),
of
1998.
models
Hosoda,
realizability,
status
models,
pp.485,
Prediction
M.S.,
using
and
of vortex
and
velocities.
collapse
in the
model,
self
for
standard
effects
agreement
velocity
performed
growth
the
(defined
flow.
solution
overshoot,
good
region-III
developed
rotational
axial
velocity,
perfectly
this
and
I, II
using
turbulence
shows
however
similarity
the
standard
of tangential
regions
periods,
a
strong
tangential
embraces
for
by
are
vortex
Since
and
model
the
simulations
axial
models.
captured
extremely
rapid
numerical
turbulent
Present
Turbulence
Approximate
an
C.J.,
Mech.
problems,
Conclusion
Chen,
order
(edited
5.
and
of
during
1981.
a
turbulent
USA,
axial
1998.
Hoffmann, E.R. and Joubart, P.N., Turbulent line
Vortices,J. Fluid. Mech. 16, pp.423-439, 1963.
Yoshizawa,A., Statistical analysis of the deviation of
the Reynolds stress from its eddy viscosity
representation,Phys. Fluids, 27, pp.1377-1387, 1984.
Pope, S.B., A more general effective viscosity
hypothesis,
10)
11)
Gatski,
J. Fluid
T.B.
stress
models
Mech.
254,
Cotton,
and
12)
Kato,
M.
and
around
Proc.
9th Symp.
Craft,
T.J.,
on
reference
Conf.
Paris,
and
J.
Fluid
of
a two
The
to
on
a strain
Refined-flow
pp.117-124,
1993.
modeling
vibrating
Turbulent
Launder,
of
of turbulent
square
ShearFlows,
B.E.
eddy
Conf.
Paris.
I. and
J.
125,
for
Suga,
cylinder,
Kyoto
Vol.1,
Extending
through
non-linear
Refined
flow
the
the
use
elements.
Modeling
and
1993.
T.,
A
prediction
Numer.
K.,
models
and
on
Hosoda,
realizability
bodies, Int.
and
viscosity
invariants
5th IAHR
Kimura,
with
IAHR
B.E.,
Measurement,
14)
5th
Development
with
Launder,
deformation
Proc.
algebraic
flows,
1993.
applicability
of
model
stationary
10-4-4-10-46,
13)
J.O.,
Measurement,
flow
explicit
turbulent
Ismail,
Proc.
and
On
1975.
1993.
and
turbulence
Modeling
pp.331-340,
C.G.,
complex
pp.59-78,
parameter,
72,
Speziale,
for
M.A.
equation
Mech.
Meth.
non-linear k-ƒÃ
of
flows
Fluids.
model
around
42,
bluff
pp.813-837,
2003.
15)
Ali,
M.S.,
Hosoda,
RANS
computations
large
scale
vortices
International
Hydraulics,
T.
of
and
Kimura,
Compound
and
secondary
Symposium
ASU,
Tempe,
on
Arizona,
I.,
channel
Unsteady
flows
currents,
with
5th
16) Lessen, M., Singh, P.J. and Paillet, F., The stability of a
trailing line vortex. Part I: Inviscid theory, J Fluid.
Mech. 63, pp.753-763, 1974.
17) Lugt, H.J., Vortex flow in nature and technology,
John-wiley & Sons, USA (ISBN 0-471-86925-2),
1983.
18) Saffman, P.G., The structure and decay of turbulent
trailing vortices, Arch. Mechanics,26(3), pp.423-439,
1974.
19) Uberoi, M.S., Mechanisms of decay of laminar and
turbulent vortices. J. Fluid. Mech. 90(2), pp.241-255,
1979.
20) Bachelor, G.K., Axial flow in trailing line vortices, J.
Fluid Mech. 20(4), pp.645-658, 1964.
21) Wygnanski,I. and Fielder, H., Some measurementsin
the self-preservingjet, J. Fluid Mech. 38, pp.577-612,
1969.
22) Devenport,W.J., Rife, M.C., Liapis, S.I., and Follin,
G.J., The structure and development of a wing-tip
vortex, J. Fluid. Mech. 312, pp.67-106, 1996.
23) Phillips, W.R.C. and Graham, J.A.H, Reynolds stress
measurement in a turbulent trailing vortex, J. Fluid
Mech. 147, pp.353-371, 1984.
Environmental
2007.
(Received: April 14, 2008)
•\ 879•\