16
On
the collision
of
drops
in
turbulent clouds
By
P.
G.
SAFFMAN
and
J.
S.
TURNER
Trinity College, Cambridge
(Received
16
November
1955)
SUMMARY
This paper proposes a theory of collisions between small drops
in a turbulent fluid which takes into account collisions between
equal drops. The drops considered are much smaller than the
small eddies of the turbulence and
so
the collision rates depend
only on the dimensions of the drops, the rate of energy dissipation
E
and the kinematic viscosity
u.
Reasons are given for believing
that the collision efficiency for nearly equal drops is unity, and the
collision rate due
to
the spatial variations of turbulent velocity is
shown to be
N=
1.30(r1
+
r2)3nln2(~/~)1/2,
valid for
r1/y2
between
one and two. A numerical integration has been performed using
this expression to show how an initially uniform distribution will
change because of collisions. An approximate calculation is then
made to take account also of collisions which occur between drops
of
different inertia because of the action of gravity and the turbulent
accelerations.
The results are applied to the case of small drops in atmospheric
clouds to test the importance of turbulence in initiating rainfall.
Estimates of
E
are made for typical conditions and these are used to
calculate the initial rates of collision, the change in mean properties
and the rate of production of large drops. It
is
concluded that
the effects of turbulence in clouds of the layer type should be small,
but that moderate amounts of turbulence in cumulus clouds could
be effective in broadening
the
drop size distribution in nearly
uniform clouds where only the spatial variations of velocity are
important. In heterogeneous clouds the collision rates are
increased, and the effects due to the inertia of the drop soon
become predominant. The effect of turbulence in causing
collisions between unequal drops becomes comparable with
that of gravity when
E
is about
2000
cm2
s~c-~.
1.
INTRODUCTION
It
has been agreed for many years that while the initial process in the
formation of clouds in the atmosphere must be one of condensation from
the vapour phase, this process is not sufficiently rapid for the small water
droplets
to
grow to raindrop sizes in the times usually available. Bergeron
(1933)
suggested that ice crystals might play a crucial part in the mechanism
Downloaded from https://www.cambridge.org/core. Caltech Library, on 20 Nov 2018 at 00:01:41, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022112056000020
On
the
collision
of
drops
in
turbulent
clouds
17
by which raindrops are formed, but observational evidence since that time
has proved that heavy rain can fall from clouds whose temperatures are
nowhere below the freezing point.
Various workers have therefore developed the idea of the growth of
raindrops by the coalescence of liquid cloud drops. These theories suppose
that drops Iarger than the mean drop size fall through the cloud under the
action of gravity, and sweep up some of
the
smaller drops in their path.
The rate of growth of the falling drops can be calculated and is found to be
a rapidly increasing function of the size of the drop.
Squires
(1952)
has investigated theoretically the size distribution of
cloud drops condensing on hygroscopic nuclei, and has concluded that
only under rather special circumstances can a few droplets considerably
larger than the mean be formed. If, for example, there are a few giant
nuclei, these will quickly grow, by condensation, to such a size that the
above coalescence mechanism will become important. Otherwise, the
calculated drop size distribution is more uniform than that observed, as
is
also clear from the work of Howell
(1949),
and the mean size is smaller rhan
that for which the above coalescence mechanism becomes more important
than condensation.
A
natural suggestion therefore is that turbulence in
a
cloud might lead
to collisions among the drops, thus giving a greater spread of cloud drop
sizes and providing the larger droplets on which raindrops could form.
(Bowen
(1950)
has indeed considered the history of drops of double the
mass of
the
rest of the drops, these drops being supposed formed by the
random collision of two drops, but the mechanism of such collisions was not
examined.) East
&
Marshall
(1954)
have discussed the previous (mostly
qualitative) suggestions which have been made about the role that turbulence
could play in precipitation, and they propose a new theory in which the
effect of random motions is regarded as being equivalent to the action of an
increased gravitational field.
It
is
concluded that turbulence could be
important in a heterogeneous cloud if the random air acceleration is com-
parable with the acceleration due to gravity.
It
should be noted that the
process pictured by these authors is still ineffective if the drops are small,
and that they predict zero collision rates in homogeneous clouds.
It
is the purpose of the present investigation to discuss a mechanism
due to turbulence which gives collisions between equal drops and which
employs
a
more realistic model of the nature of the turbulent motion than
that used by East
&
Marshall. The collision frequencies are found
in
terms of the rate of energy dissipation per unit mass in the cloud and this
quantity is estimated from the large scale properties of the air motion inside
the clouds.
2.
THE
NATURE
OF
THE
TURBULENT MOTION
East
&
Marshall regarded the turbulence as equivalent in its effects to
a
random motion in time of the
whole
air parcel containing all the drops,
and neglected the spatial variations which are surely an essential feature of
F.N.
B
Downloaded from https://www.cambridge.org/core. Caltech Library, on 20 Nov 2018 at 00:01:41, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022112056000020
18
turbulent motion. All collisions on their model are therefore due to the
different motion, relative to the air, of drops of different sizes; and hence
they found that equal drops did not collide. The effect of the spatial
variations, however, is
to
give neighbouring drops different velocities and
thus cause collisions, whatever the ratio of the sizes of the drops.
The cloud droplets that we are considering are usually smaller by at
least
an
order
of
magnitude than the length scale of the small eddies
of
the
turbulence, and
so
the relative motion of two neighbouring drops will be
governed by the small scale motion.
It
has been pointed out by Batchelor
(1950) and others that because the Reynolds number for turbulent motion
in the atmosphere is usually large, the similarity theory of turbulent motion
will hold for the small scale motion. This theory implies that for scales
of motion sufficiently small compared to the energy-containing eddies, the
motion is isotropic and the mean values of quantities related to the turbulence
will depend only on the kinematic viscosity
Y
and the rate
of
energy dissipation
per unit mass
E,
provided that
the
quantities concerned depend strongly on
the small scale properties of the turbulence. For example, the
relative
diffusion of a cloud of smoke can be treated in this way, whereas diffusion
from a fixed source can not, since in the latter case
the
large eddies are also
important.
It
follows that the effect
of
turbulence in causing collisions
between neighbouring droplets will also depend on the rate of turbulent
energy dissipation per unit mass
E,
and it will be necessary to make an
estimate of this quantity in clouds under various conditions. We shall
do
this first before describing the mechanism of collisions.
Brunt (1939) has given as an average value in the lowest few kilometres
of the atmosphere
E
=
5
cm2 Few direct measurements which allow
E
to be estimated in clouds have been made, but
R.
J.
Taylor (1952) has
deduced values of
c
of
the
order of 1000cm2sec-3 close to the ground in
moderate winds. Another approach to
the
problem is available however.
The results of laboratory experiments (e.g. Batchelor
&
Townsend
1948) have indicated that the rate of turbulent energy dissipation is usually
of order
u3/l,
where
u
is a root-mean-square turbulent velocity and
1
is a
length scale associated with the energy-containing eddies. Measurements
of accelerations experienced by aircraft in bumpy conditions indicate that
fluctuating velocities of a few metres per second are common, with corre-
sponding eddy sizes of the order of tens or hundreds of metres. If we take
as typical figures,
u
=
2
metres/sec,
I=
50
metres, we obtain€= 1600 cm2
We might suppose that something less than the mean figure
of
E
=
5 cm2 sec-3
would be applicable to stratiform clouds where there is a small mean
velocity, and take a larger value say
E
=
1000 cm2 as an estimate of the
conditions in turbulent cumulus clouds. However,
it
may be possible to
make better estimates than these for the different types of clouds.
P.
G.
s.fJ.1Mt
adJ.
S. Turner
3.
DISCUSSION
OF
THE COLLISION
PROCESS
There are two ways in which turbulence causes collisions between
Firstly, there are the spatial variations of the
neighbouring droplets.
Downloaded from https://www.cambridge.org/core. Caltech Library, on 20 Nov 2018 at 00:01:41, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022112056000020
On
the
collision
of
drops
in
turbulent clouds
19
turbulent motion referred to previously. Collisions due to this process
can conveniently be called ‘collisions due to
the
motion
of
the droplets
with the air’. Secondly, each droplet is moving relative to the air sur-
rounding it, owing to the fact that the inertia of a droplet is different from that
of an equal volume of air.
It
follows that neighbouring droplets of
unequal
size will have different velocities (since the inertia of a droplet depends
on its size), and this also will lead to collisions. This process is called
‘
collisions due to the motion relative to the air
’.
The latter process will
not give collisions between droplets
of
equal sizes, such collisions being
due to the first process.
Before we can calculate the rate
at
which collisions between droplets
occur, it
is
necessary to consider the effect that the presence of a droplet
has on the motion
of
neighbouring droplets, that is, we must consider the
distortion of the flow due to the presence of a drop.
A
measure of this
distortion is the collision efficiency, which can be defined as that proporiion
of drops which would have collided in the absence of distortion, actually
to
do
so.
It
is clear that the collision efficiency must be dependent on the
nature
of
the flow. Collision efficiencies were calculated by Langmuir
(1948) for the case
of
small drops suspended in the steady laminar flow
around
a
fixed large sphere. His numerical results are only likely to be
accurate in cases where the basic assumptions are satisfied, although they
have been extensively applied beyond their range of validity by others, and
by East
&
Marshall in particular.
It
is difficult
to
justify the application
of Langmuir’s results to the present problem, since we are concerned with
a case in which the colliding drops are of nearly equal sizes. Recent
measurements, by Telford, Thorndike
&
Bowen (19SS), of collection
efficiencies for nearly equal drops somewhat larger than cloud drop sizes,
have indeed given values considerably higher than would be predicted by
an application of Langmuir’s theory.
Now the Reynolds number of the relative motion of two nearly equal
approaching cloud droplets will usually be much less than one, and some
relevant experimental evidence for this range of Reynolds numbers is
provided by the experiments of Manley
&
Mason
(1952,
1955). They
observed the collisions between glass spheres and between air bubbles sus-
pended in a uniformly sheared viscous liquid, and showed that in these
circumstances, the collision efficiency is indeed unity, that
is,
the distortion
of
the flow does not influence the collision rate.
It
thus seems that, in the absence of further evidence, it is not unreason-
able to take the collision efficiency
of
nearly equal droplets as unity
;
this is
equivalent
to
neglecting altogether the distortion of the flow by a drop.
We are interested primarily in collisions between nearly equal drops,
and as a beginning we shall confine our attention to collisions due to motion
with the air. Collisions due to motion relative to the air are not unimportant,
but this process does not give collisions between equal drops. Later, we
shall make an estimate of the relative importance of the two processes.
It
is useful to keep in mind the qualitative picture
of
the local shearing
motions which lead to collisions between drops carried along with the
flu$.
B2
Downloaded from https://www.cambridge.org/core. Caltech Library, on 20 Nov 2018 at 00:01:41, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022112056000020
20
P.
G.
Saffman
adJ.
S.
Turner
For two close points in a turbulent fluid, the relative motion is that of
uniform strain. Taking an origin at the centre
of
one drop and the
coordinate axes in the directions of the principal rates of strain, the stream-
lines of the relative motion in one of the coordinate planes are as shown
in figure
1.
The other drops are moving with the air along these stream-
lines and since we neglect the distortion of the flow field due
to
the presence
of the drops, for the reasons mentioned above, the collision rate of the
‘fixed drop’ with other drops is just the flux
of
fluid inwards across the
surface of a sphere, concentric with the fixed drop and
of
radius equal
to
the
sum of the radii
of
the two approaching drops, multiplied by the number
density
of
the other drops. The equivalent calculation for a uniform
laminar flow assumes,
of
course, that all drops in a cylinder parallel to the
flow and containing the effective cross-section
of
the fixed drop should meet
that drop.
Y
-x
-
-Y
Figure
1.
Streamlines
of
the relative motion in one
of
the principal planes.
4.
COLL~SIONS
BETWEEN
DROPS
MOVING
WITH
THE
AIR
We now proceed to the calculation of this collision rate. Let the mean
concentrations
of
two sizes of drop in a given population be
n,
and
nz
per
unit volume, and their radii
rl
and
r2
respectively. The collision radius
for a pair
of
drops, one of each type, will be just the sum
of
the two radii,
Downloaded from https://www.cambridge.org/core. Caltech Library, on 20 Nov 2018 at 00:01:41, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022112056000020