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115 TOWARDS BETTER UNDERSTANDING OF PREFERENTIAL CONCENTRATION IN CLOUDS:
DROPLETS IN SMALL VORTICES
K. Karpinska 1∗
, S. P. Malinowski1
1Institute of Geophysics, Faculty of Physics, University of Warsaw, Warsaw, Poland
1 INTRODUCTION
Recent studies attribute the evolution of droplet size
distribution in warm convective clouds to enhance-
ment of collision-coalescence by turbulence (see
e.g. Devenish et al. (2012), Shaw (2003)). One of
the influences turbulence has on droplets collision-
coalescence is its effect on droplet positions lead-
ing to their uneven distribution in space. The aim of
this study is a better understanding of the preferenial
concentration of droplets from analytical and numer-
ical analysis of droplet motion in vortex tubes: small
coherent structures characteristic for high Reynolds
number turbulent flows. Former research of such ef-
fects by Hill (Hill, 2005) and by Markowicz (Markowicz
et al., 2000) was limited to horizontally oriented vor-
tex tubes only. Herein we analyse tubes which are
parallel or oblique to the direction of gravity.
2 VORTEX TUBE MODEL
Line vortex is a theoretical model of 3d structure of
constant circulation (Γ) and singular vorticity concen-
trated on a straight line. In cylindrical coordinates
(r, φ, z)in which vortex singularity lies on Z axis, its
vorticity is given by:
~ω =Lw
δ(r)
rˆez(1)
while Lw=Γ
2πis a parameter of vortex circulation.
We use constant velocity field generated by line vor-
tex with stretching of strenght γ[1
s]as a model of
vortex tube:
~va=−γ
2rˆer+Lw
rˆeφ+γzˆez(2)
Numerical simulations were done for visualiza-
tion purposes with parameters corresponding to
cloud/water droplets in an airflow. Two sets of vortex
parameters were chosen for numerical simulations:
•strong vortex: γ= 30 1
s,Lw= 0.025 m2
s,
∗Corresponding author: Pasteura 7, 02–093 Warsaw, Poland,
e–mail: karpinska@igf.fuw.edu.pl
•weak vortex: γ= 0.51
s,Lw= 2.5·10−4m2
s.
Values of other parameters used were as follows:
air kinematic viscosity µ= 1.776 ·10−5kg
ms , water
density ρw= 1000 k g
m3, gravity of Earth g= 9,81 m
s2.
We arranged our vortex model to cover all possible
orientations with respect to gravity direction by intro-
ducing an angle θ∈[0, π]between gravity vector and
vortex axis (see Figure 1):
~g =−g(sin θˆey+ cos θˆez)
Figure 1: Scheme of gravity vector orientation in re-
spect to the vortex axis
3 EQUATIONS GOVERNING DROPLET MOTION
We assumed that droplet is a point particle and its
motion in fluid is determined by viscosity and gravity
forces only, so no other hydrodynamical forces and no
interaction with other droplets were included. Stokes
equation with gravity was used as droplet equation
of motion. In a fluid flow with velocity field va, for
a droplet of mass min a position ~r with inertial re-
sponse time τand under gravity force Fg=m~g this
equation is expressed by:
m¨
~r =1
τm(~va−˙
~r) + m~g. (3)
Equation (3) was nondimensionalized with use of τ
as time scale and length scale S=√Lwτconnected
1
to vortex circulation. The resulting equations of mo-
tion in a plane perpendicular to the vortex axis (here
in (r, φ)coordinates) separate from motion along Z
axis.
¨r−r˙
φ2=−(K1
2r+ ˙r+K3sin(φ))
2 ˙r˙
φ+r¨
φ=1
r−r˙
φ−K3cos(φ))
¨z=K1z−˙z−K2
.(4)
Motion of a droplet in case of gravity parallel to the
vortex axis (θ= 0) depends on two following dimen-
sionless parameters:
L1=γτ,
L2=gτ 2
S.(5)
and on three in case with gravity direction nonpar-
allel to the vortex axis (θ6= 0):
K1=γτ =L1,
K2=gτ 2
Scosθ =L2cosθ,
K3=gτ 2
Ssinθ =L2sinθ.
(6)
L2parameter has a direct physical interpretation
as a rate of droplet terminal velocity (gτ ) to charac-
teristic velocity ( S
τ) and it determines whether motion
of droplet is mainly gravitational or circular. L2is in-
dependent of vortex stretching, whereas L1express
rate of velocity increment due to stretching at droplet
characteristic length (γS ) to the droplet characteristic
velocity ( S
τ). K2and K3parameters has obviously
the same interpretation as L2, but their impact on so-
lutions change with θangle.
4 DROPLET DYNAMICS AROUND LINE VORTEX
We described droplet motion in general by analyti-
cal calculations and also by numerical simulations in
cases of weak vortex and strong vortex.
4.1 Motion in direction parallel to the vortex axis
The third equation in (4) was solved with initial con-
ditions z(0) = z0,˙z(0) = 0. The following formula
describes motion of droplet along Z axis (vortex axis):
z(t) = z0−z0b
λ+−λ−
[λ+exp(λ−t)−λ−exp(λ+t)] + z0b,
(7)
λ=λ(K1), λ+>0, λ−<0(8)
while z0b=K2
K1S.
Direction of motion along Z axis is therefore de-
termined only by the initial position of droplet z0. If
z0=z0b(z0b>0) the droplet stays in unstable
steady position in respect to this motion. Stable posi-
tion is proportional to second power of droplet radius
and decreases with growing θangle and stretching
strenght: z0b∝γ−1R2cos θso for droplets of dif-
ferent sizes their stable positions get closer to each
other with growing θand γ. If z0> z0bor z0< z0b
droplet moves nearly exponentially up or down re-
spectively along the vortex axis. Its linear accelera-
tion equals −gcos θat the start, then decrease due to
viscosity force and then rapidly increase due to vortex
stretching.
These formula and conclusions are valid for the
whole range of angles θ∈[0,π
2]. In a special case of
θ= 0 general parameters of motion K1,K2become
already mentioned L1,L2parameters.
4.2 Motion in plane perpendicular to the vortex
axis
Motion in a plane perpendicular to the vortex axis
shows strong qualitative dependence on angle θ. For
this reason cases of “vertical vortex”, meaning θ= 0
and “oblique vortex” with θ6= 0 are analysed sepa-
rately.
4.2.1 Line vortex parallel to gravity
In this case there is only one kind of solution for
motion in plane (r, φ): every droplet has its circular
stable, periodic orbit on which radial viscous force
and centrifugal force equalize. Radius of this orbit
is rorb =4
q2τ
γ(Γ
2π)2so it increase with increasing
vortex circulation and droplet radius as well as with
decreasing stretching strength. Stability of the orbit
guarantees that trajectories of all the droplets spirals
into it in finite time. Angular velocity of initially station-
ary droplets very quickly increases and decreases
at the beginning of the simulation and later tends to
smaller, constant value of ˙
φorb =pγ
2τ.
Figure 2 presents a 3d droplet trajectory with initial
position r0> rorb, z0< z0band initial zero velocity.
If time that droplet needs to get on its 2d steady orbit
is small in comparison with timescale of motion along
Z axis than it can reside on its 3d orbit for significant
amount of time.
Numerical simulations for 3d droplet motion in the
weak vortex for radii range 1−20µm gave the follow-
ing results:
•There is significant difference of "‘residence
time"’ of small and large particles: time of get-
ting on the steady orbit for small droplets is very
short in comparison to characteristic time of ver-
tical fall (or lifting) resulting in the effect of long
stay of small droplets on their circular, steady,
periodic orbits.
2
Figure 2: Trajectory of droplet of radius R= 10µm in
weak vortex with initial position r0> rorb,z0< z0bin
time t∈[0,10s].
•Trajectories of droplets in the same initial posi-
tion but different initial velocities (zero and fluid
velocity) show strong dependence on this initial
condition, which can be described as a sling ef-
fect (Falkovich and Pumir, 2007).
4.2.2 Line vortex nonparallel to gravity
Gravity influence destroys axial symmetry of motion
in a plane perpendicular to the vortex axis. It mani-
fests in the equations by additional K3parameter de-
pendence. In consequence solution of equations in a
form of round, stable, periodic orbit does not exist for
any droplet in this case. Gravity influence on motion
results however in possibility of apperance of equillib-
rium points in the IV quadrant of the plane described
with (x,y) coordinates. Positions of these points de-
scribed in (r, φ)are as follows:
rst±=√2K3
K1r1±q1−K1
K2
3
2
φst±=−arcsin(1
√2r1±q1−K1
K2
3
2)
(9)
under the condition K2
3≥K1. This condition splits
into two:
L2
2≥L1,
θ∈(arcsin(qL1
L2
2
),π
2).(10)
The existence condition was pictured in Figure 3:
for given vortex of γand Lwequillibrium point may
exist only for droplets of radii above the drawn sur-
face. Droplets which fulfill the first part of condition
Figure 3: Boundary surface presenting equillibrium
points existence condition for a range of droplets
reaching 100 µm.
(10) with equality (are positioned on the surface plot-
ted in Figure ??) for a given γand Lwcan have an
eqillibrium point only when the vortex axis is perpen-
dicular to gravity vector, θ= 0. For example the
boundary droplet radius stemming from above con-
dition for strong vortex is 84.07 µm, for weak vortex it
is 9.56 µm.
Linear stability of the two solutions for equillibrium
points (see equation (9)) was examined (as in Marcu
et al. (1995)) for the case of Burgers vortex) and the
conclusions are as follows:
•rst−is always unstable,
•rst+is unstable only under conditions: K1∈
(1
4,1
2)and K2
3≤K2
1−1
2K1+1
8
1
2−K1.
Numerical simulations show also apperance of
noncircular limit cycle under certain vortex/droplet
conditions. It can be a unique stable solution or com-
pete with the stable equillibrium point. Generally the
result of leaning the line vortex with respect to gravity
leads to droplet 2d motion in which it approaches one
of two types of attractors: either a stable limit cycle or
a stable equillibrium point. Figure 4 presents exam-
ples of trajectories of same-sized droplets in the weak
and the strong vortices accordingly in which we can
see the situation of coexistence of noncircular stable
limit cycle with stable eqillibrium point.
In the weak vortex we observe significant influence
of gravity on motion of droplets only for those of radii
close to boundary radius. For smaller droplets there
are stable limit cycles of shape close to circular. For
bigger droplets, as in Figure 4a) fast approaching the
limit cycle was observed with its shape slighly devi-
ated from circular in closeness of stationary point.
3
(a) R= 10 µm, weak vortex
(b) R= 85 µm, strong vortex
Figure 4: Trajectories of 36 droplets seen in XY
cross-section of radius R distributed uniformly in a
plane z= 0 on a rectangle l=8cm in a vortex with
θ= 0.45π.
In the strong vortex however this influence is easily
seen even for droplets much smaller than boundary
radius. The shape of trajectories generally is very
complicated, there are also great fluctuations of ve-
locity while attracted by the stable limit cycle. This
is shown for droplets of radius close to boundary in
Figure 4b).
4.3 Various size droplets motion simulations
Figures 5 and 7 are frames from 3d simulations of
motion of various size droplets in the weak and in the
strong vortex. They are cross-sections perdpendicu-
lar to the vortex axis while Figure 6 is a projection of
3d picture of the same visualization for weak vortex.
Red and orange lines are plots of equillibrium points
positions for those droplets from a chosen range for
which they exist. The overlaying of these lines by
endpoints of trajectories of droplets is a visualization
for good agreement between analytical and numeri-
cal results.
Different types of droplets behaviour (periodic or-
bits, limit cycles, equillibrium points) described above
strongly influence space distribution of different size
droplets as seen in Figures 5, 7 and 6. This effect is
strong especially if the timescale of motion along the
vortex axis is increased by vortex leaning. Figure 6
indicates that in oblique, line vortex droplets of vari-
ous radii tend to separate in space. Smaller droplets
are attracted by their periodic orbits around the vor-
tex axis while motion of the bigger ones is determined
more by gravity and equillibrium point attraction.
5 CONCLUSIONS
Features such as stable periodic orbits, stable eqil-
librium points and limit cycles were identified qual-
itatively as three-dimensional structures that may
lead to enhancement of preferential concentration of
droplets in clouds. Conditions for existence of pe-
riodic orbits and equillibrium points were derived as
well as their stability was verified. Numerical solu-
tions agree with analytical results.
Acknowledgements
This research was supported by the Pol-
ish National Science Centre with the grant
2013/08/A/ST10/00291.
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4
(a) t=0s (b) t=1s
(c) t=3.2s (d) t=15s
Figure 5: Positions of 100 droplets seen in XY cross-
section of various radius from range [1, 20 µm] start-
ing in a plane z= 0 on a rectangle l=8cm, in the weak
vortex with θ= 0.45π.
Figure 6: Positions of 100 droplets of various radius
from range [1, 20 µm] starting in a plane z= 0 on a
rectangle l=8cm, in the weak vortex with θ= 0.45πin
t=4.5 s.
(a) t=0s (b) t=0.1s
(c) t=0.2s (d) t=1s
Figure 7: Positions of 100 droplets seen in XY
crossection of various radius from range [75, 95 µm]
starting in a plane z= 0 on a rectangle l=8cm, in the
strong vortex with θ= 0.45π.
5
