Content uploaded by Szymon P. Malinowski

Author content

All content in this area was uploaded by Szymon P. Malinowski on Oct 26, 2014

Content may be subject to copyright.

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.

References

Devenish, B. J., and Coauthors, 2012: Review article.

droplet growth in warm turbulent clouds. Q. J. R.

Meteorol. Soc.,138, 1401–1429, doi:

.

Falkovich, G., and A. Pumir, 2007: Sling ef-

fect in collisions of water droplets in turbulent

clouds. J. Atmos. Sci.,64, 4497–4505, doi:

.

Hill, R. J., 2005: Geometric collision rates and trajec-

tories of cloud droplets falling into a burgers vortex.

Phys. Fluids,17, 037 103, doi: .

Marcu, B., E. Meiburg, and P. K. Newton, 1995: Dy-

namics of heavy particles in a burgers vortex. Phys.

Fluids,7, 400–410, doi: .

Markowicz, K. P., K. Bajer, and S. P. Malinowski,

2000: Inﬂuence of the small-scale turbulence

structure on the concentration of cloud droplets.

Proc. of 13th Conf. on Clouds and Precip., IAMAP.

Shaw, R. A., 2003: Particle-turbulence interac-

tions in atmospheric clouds. Annu. Rev. Fluid

Mech.,35, 183–227, doi:

.

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