Stretching of a vortical structure: filaments of vorticity
Stretching of a vortical structure: filaments of vorticity'
ortical structures are commonly formed in flows: One has Vnoticed a bath tub vortex, or a vortex formed with a kitchen mixer. The largest vortical structures can be observed in Earth's high-atmospheric flows where cyclones or anti-cyclones can reach up to 1000 km diameter with typical velocities of the order of 100 kmlh. Large structures can also be seen in oceans with typical diameters of 100 km with velocities of few tens of kmJh. On the other side of the spectrum, the smallest vortices also have a limit, which is not due to the size of the planet, but to viscosity: below a certain size, viscous dissipation does not allow for an organized coherent structure. Its whole energy is dissipated as heat in the flow. On an intermediate scale,
tornados are spectacular A Fig. 1: Picture of a tornado.
structures or vortices (Fig. I).
Intituively, one Wlderstands that a vortex can be described by
two parameters: Its size, and its circumferential velocity. The size of
the core of the vortex is defined as the distance ro from the vortex
axis where the azimuthal velocity Vii is maximal Vii(ro) = V9max. For
distances larger than ro, the rotationel velocity decreases and goes
to zero at "infinity". Below this diameter 2To, the vortex is approxi
mately in a state of solid-body rotation, as shown in Figure 2a.
Another important parameter is the vorticity mwhich is the
curl of the velocity m =TotV. Alternatively, it can be written as
twice the angular velocity n of the vortex. In the case represented
in Figure 2a, the vorticity has only a non-zero component in the
direction along the vortex axis. This vorticity is represented in
• Fig.2a: Example of
azimuhtal velocity of a
vortex Ve (r).
• Fig. 2b: Vorticity
m(r) and pressure (P-Po)
profile of the velocity
profile (Fig. 2a).
iL ..._..._._. _....__..
In this example, one observes indeed that the vorticity is con
centrated in the vortex core. A flow with Vii - 1fT outside the core
has been chosen. This gives a zero vorticity outside this region.
Note that such a velocity field depends on r although its axial vor
ticityis zero: Wz = liT [%r(Ve!r)T-Tdv,:tOO] =O. In a similar way,
"a non-zero vorticity does not mean that a vortex is present: A
shear flow such as a bOWldary layer flow on a flat wall can be sta
ble Wltillarge Reynolds number (Re= UL/ v). For all vortical flows,
vorticity is a very important parameter since it locates vortices
and gives their "intensity".
An important mechanism that enhances the vorticity is the
stretching. Stretching a vortex along its axis will make it rotate
faster and decrease its diameter in order to maintain its kinetic
momentum constant. This is analogous to the kinetic momentum
conservation law in solid mechanics. A well known example is the
ice-skater who turns faster as she brings her hands near her body,
and vice versa. An example in fluid mechanics is the bath-tub vor
tex that rotates faster and becomes smaller as it goes from the fluid
surface to the exit. More precisely, a stretching ris an acceleration
of the axial velocity along the vortex (r= VV). The vorticity equa
tion is obtained from the curl of the Navier-Stokes equation:
om/dt+(V. V) 00 = (00. V) V + v~m.
The first term represents the temporal evolution of the vortici
ty, the second is an advection term, the third is the stretching term
and the fourth represents the viscous effects. When stretching is
parallel and in the same direction as the vorticity, the term (m. V)
is positive and amplifies the vorticity (om/ot). The vorticity
increases and the viscous term (v~m) becomes large enough to
exactly coWlter balance the amplification term (0). V)V.An equi
librium is then reached which imposes the diameter of the vortex
r m= vcolT02, hence TO "" (v/i) 1/2. From the general perspective of
view, the stretching of the vorticity in the flow leads to its amplifi
cation and its confinement. Note that it is not the vorticity which
is conserved, since it is amplified by stretching, but the circula
tion r=oJcU.dl. Circulation aroWld a curve C is the total vorticity
inside this curve.
Vortex stretching is a very important mechanism in fluid
dynamics: it usually corresponds to the presence of vortical struc
tures which are much more intense that those produced by simple
shear or rotating flows. In particular, it is now well known that
local stretching of vorticity in turbulent flows produces very
intense vortices called "filaments of vorticity". A large part of the
scientific community working on turbulence believes that these
structures are very important in the dynamics of turbulent flows,
although their structure, their dynamics or their instabilities are
not fully Wlderstood. The study of these structures of intense vor
ticity is hence extremely important for both fundamental research
and applied science (flow
control, ... ).
Two experiments have
b) been built to produce fila
ments without turbulent
flows. Only the fundamental
ingredients have been kept,
i.e. initial vorticity and
ro r stretching. These set-ups
allow the study of the fila
ments of vorticity with a
control over various
parameters of the experiments.
In that way, an isolate
"standard" vortex is produced for a study of its structure, its
dynamics, its instabilities and its turbulent burst.
The two experiments allow a direct study of the mechanisms
behind a vortex structure along with its dynamics. Note that the
vortex structure is much more complex than models such as
The first experiment corresponds to the stretching of the vor
ticity of a laminar boundary layer flow in a water channel. The
second produces a stretched vortex between two co-rotating
discs. In the latter experiment, filaments of vorticity of stronger
intensity are produced as the injected vorticity is much larger with
rotating discs than with a boundary layer flow. These two experi
ments cover a large range of vorticity.
Experiment in a water channel (stretching of a boundary
A vortex is generated by stretching the vorticity of a stable lami
nar boundary layer flow which, under natural conditions, does
not produce a vortex. The initial vorticity sheet occurs in the lam
inar boundary layer (Reo"" 100). Figure 3 shows a sketch of the
channel section where the study is performed. On the right hand
side of the figure; the flow develops boundary layers on each wall.
The boundary layer on the lower plate has, for instance, a non
zero component of the vorticity (autay:t- 0). This "initial"
vorticity (0; will be artificially enhanced to a powerful single vor
tex. To enable this, a stretching, along this initial vorticity is
carried out through suction from slots located in lateral walls. The
suction generate a stretching, i.e. an acceleration or a transversal
velocity gradient VV, parallel to the initial vorticity (0;. When the
stretching is large enough, a strong vortex is produced between
the two suction slots.
Re. = 100
... Fig. 3: Experimental set-up.
As the mean flow is laminar and the velocities are slow (0 to 10
cm/s), the study of a stretched vortex becomes easier compared to
a study in a turbulent flow. Some examples of flow visualization
are shown in photographs of Figures 4, which were obtained from
either the injection of fluorescent dye in conjunction with argon
laser lighting or from the injection of food colour dye upstream.
If the whole flow goes through the suction, a permanent vor
tex is produced. In that case, its structure is studied with velocity
measurements as well as its eventual instabilities. If only a part of
the flow goes through the suction slots while the rest goes down
stream (on the left hand side of the Figure 3), the vortex tends to
be advected by the flow while the suction tends to keep it attached.
Depending on the ratio between these two mechanisms, the vor
tex can remain attached or can explode as a turbulent spot. The
vortex cannot persist after detachment from suction slots since it
lost its stretching, leading to an interruption of the axial flow. It
will therefore break up. Another vortex is produced that follows
the same dynamics and so on with a very well defined frequency.
Mechanisms behind the formation of vortices have been deter
mined through visualizations. It has been observed that vorticity
filaments were generated by the roll-up of a fluid sheet around the
~ Fig.4a: Visualization of
the cross-section of a
stretched vortex (plan x,y
on FiglKe 3).A sheet of
fluoresce in is injected just
before stretching. Vortex
formation by roll-up of '
fluid sheets on themselves l,'
is observed (image size:
3 cm, vortex core diameter: I
0.3 cm). j
An important point that has been studied is the structure of the
vortex (Fig. 4). Indeed, the localization of the stretching does not
allow a simple model for this vortex, such as what has been pro
posed when the stretching is uniform (Burgers' vortex). The axial
component of the velocity must depend on the distance to the
vortex axis r (Fig. 4b). A model will be proposed very soon.
... Flg.4b: Visualization of a vortex by injection of two small dye
jets.Top view (plan x,z of Figure 3;. Flow comes from the bottom of
the image towards the top and is stretched through suction
represented as two small rectangles on the image (size of the
image: 12 cm).
PIV technique (Particles Image Velocimetry) was used for
measurements of the velocity field in a transverse cross section of
the vortex (r, ()). This technique consists of seeding the flow with
small particles that are traced for the determination of the
velocity fields. Two successive images of the 2D cross-section of the
flow are recorded with a video camera. These images are usually
obtained thanks to a laser sheet illuminating only particles
within its light sheet. The velocity field is then evaluated from the two
successive images by following the displacements of particles.
Figures 5 gives an example of what can be obtained with this
<C Fig. ~~~------I
Visualization of a
vortex by injection
of two small dye
jets. Front view
(plan x,y of Figure
3;. Flow comes
from the right of
the image (size of
the image: 4 cm).
Thanks to different experimental techniques available in our
laboratory, we were able to measure quantitatively the evolution of
the vortex characteristics as a function of the stretching y. Main
results are summarized in the following figures:
Figure 6a plots the amplification of the initial vorticity 0Jt/CO; as
a function of the stretching. The initial vorticity CO; was deduced
from the velocity profile just before stretching and the final vor
ticity 0Jt is evaluated from the velocity profile of the vortex. This
experiment enables a large range of parameters to be explored as
the vorticity can be enhanced by a factor of lOO! Figure 6b shows
the maximum azimuthal velocity Vomax and the mean radius of
the vortex To as a function of the stretching. The results presented
in this figure are obtained from PIV measurements technique.
One can observe that To strongly depends on the stretching and is
not fixed by the diameter of the suction hole. It follows that
To cc (v/ y) 1/2 which is explained by the balance between the
amplification of vorticity term «(j)V) V and the viscous dissipation
term V~(j) in the equation of the vorticity.
In summary, in this experiment we can generate a vortex with
characteristics of VlImax '" 1 cm/s and 2To '" 1 cm for a stretching
just large enough to produce a vortex. For maximum stretching,
just before the vortex becomes too unstable and explodes in a
20 • •
0-"...__· _, -_ _- .
.. Fig. sa: (left) Velocity field obtained by PIV on a cross-section
of the vortex (plan x,y). Fig- 5b: (centfe)Velocity modulus that is
approximately the azimuthal velocity VII the radial velocity Vr
being much smaller than V/I.The shape of the vortex (its
ellipticity) can be determined with these measurements. Red color
coresponds to the largest velocity (VII max) that is reached for '''''0,
i .while blue color correspond to the slowest velocity. Modulus of
I velocity profiles are shown on the sides of the image, with
; maxima at '''''0 for Y"'Y/I mox and a minimum at the center of the
f vortex where radial and azimuthal velocities are almost zero.
! Eig..--?£ (right)The curl of the velocity gives the axial vorticity.
, Particularly, one observes that vorticity is concentrated in the
I vortex core, and that the flow is quasi irrotational around. This
! calculation has been performed after a Gaussian-type filtering of
I the ~:city fiel~._of_F_ig_U_re_5_a_.
bulent burst, the vortex characteristics become as large as
VlImax '" 20 cmls with a diameter 2To = 0.2 cm.
Indeed, we observe that the more the vortex is stretched, the
mpre its vorticity is amplified, the more its diameter decreases,
and the faster it rotates; the circulation being the only quantity
that is conserved. However, note that in. this experiment, the cir
culation increases, but for another reason: when stretching is
increased by increasing the flow rate of the suction, entrance
conditions are modified as well, and the injected initial vorticity
becomes more important.
When the flow rate downstream is not zero, a different
dynamics is observed. In that case, the flow tends to advect the
vortex. Above a threshold, the vortex is unstable and explodes as a
turbulent burst. Mechanisms leading to this turbulent explosion
are very complex and depend on different kind of instabilities.
Experiment between two co-rorating discs
Another experiment devoted to the study of stretched vortices
consists of stretching the vorticity of a flow between two co
rotating discs: The discs of diameter 2R (5, la, or 15 cm) are
positioned in front of each other with a gap d (0 < d < 20 cm)
between the disc surfaces. The discs can rotate at identical speeds
.. Fig. 7: Experimental set-up.
~ fig,.Ji Visualization
of a vortex.The diameter
of the discs is here 10cm.
Q (0 < Q < 1200 rpm). At the centre of each disc, a 0.5 cm diame
ter hole allows for suction thanks to a pressure drop (see Fig. 7).
This set-up is placed in a water tank whose walls are at least 20
cm away. A vortex is then produced through the vorticity inject
ed by the rotation of the discs and enhanced by stretching (Fig. 8).
Unlike the experiments of the straight channel, the two main
control parameters (rotation and stretching) are independent
here. Furthermore, the injected vorticity is much larger than in
the experiment of the straight channel. These two experiments
represent then a very large range of situations.
Pressure is an important quantity that is very difficult to mea
sure in a vortex. We have shown that a probe, even very small
(200 ~ diameter) strongly destabilize the vortex. The mean pres
sure in the vortex has been deduced from the measurements of
the feedback of the rotation on the stretching: As illustrated in
Figure 8, the stretching is produced by a suction system based on
a pressure drop between the water level in the big tank (that is
maintained at a constant level), and the water level of a small
tank connected to the suction slots. The flow rate is measured
when the discs are at rest and there is no vortex. A vortex is gen
erated a few seconds after the discs start to rotate with the
conditions that the stretching, the distance between the discs, the
rotation speed etc are suitable. One observes a flowrate drop iIl the
small tank. This is explained by the presence of the vortex which
induces a pressure drop in its core. This pressure drop counter
balances the suction and the stretching, hence the flowrate of the
small tank. From the flow rate drop, we evaluate the mean pres
sure drop in the vortex core: The flow rate with a vortex and a
difference of water-level hI between the two tanks, is the same as
that without a vortex but with a smaller difference of water-level
0.5 I 1.5
B = ( P Q2 R' I p. ) III
... Fig. 9: Pressure drop into the vortex P/Powhere the initia(', ,c '"
pressure drop Po is imposed by the difference between the w~te;""":
level in the two tanks when there is no vortex (0=0) as a funciron;"
of a parameter 8 that represents the ratio between the pressure:;:':';';
drop generated by the rotation and Po. Saturation proves thatli:"~.'! ~
above a threshold, rotationJost its efficiency. .::l!
kr·/~;;::.. E.i •1Oa: Long
bubble into the vortex.
.. Fig.10b: Chain of
h2. The pressure drop P in the vortex core is given by the relation
Pv = pgLlh, where ilh = hI - h2. This pressure drop is measured
as a function of the initial stretching (i.e. hI), the size R of the
discs, the distance d between the discs and the rotation velocity
Q (Fig. 9). This study has been performed with F. Moisy and B.
The pressure drop in the vortex can be large enough to trap
micro bubbles naturally present in the water of the tank. This can
lead to the formation of a long bubble that stays along the vortex
core (Fig. lOa). The bubble is observed to break up into regularly
spaced smaller bubbles along the vortex axis (Fig. lOb). This break
up is due to the Rayleigh instability.
An interesting point is that, even if the flow rate in the suction
holes is of the order of a few lImin, with an axial velocity of a few
m/s, the pressure drop in the vortex core is large enough to
maintain the bubbles. .
Note that it is not cavitation that is observed, as it would require
the pressure in the vortex core to be smaller than the vapor pres
sure of the water. The observations show only concentration of
micro-bubbles naturally disolved in water.
The study of stretched vortices is a challenging research topic
that presents lots of experimental difficulties due to large velocity
gradients, three dimensionality and unsteadyness. However, we
believe that mechanisms studied in these "standard" vortices are
the same as those of filaments of vorticity in turbulent flows. We
hope in that way to contribute to a better understanding and
modeling of turbulence, with flow control and prediction as our
main objectives. Flow control is indeed an important challenge for
the future, in order to reduce the energy losses in turbulent flows
(drag reduction, better efficiency in mixing, ... ). Prediction is also
an important challenge for instance in flow control (better pre
diction leads to better control), or in weather forecasting.
I This article is based on an original version published in the Bul
letin de la SFP (French Physical Society), 132, pA, I dec.2001
2 A boundary layer is a thin layer that develops over a wall and
connects the wall where the velocity is zero and the free-stream
flow. It is inside this layer that viscosity plays a role whereas it has
a much smaller effect in the rest of the flow. This region corre
sponds to the zone where the velocity gradient produces vorticty.
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