Conference PaperPDF Available

Towards better understanding of preferential concentration in clouds: droplets in small vortices.

Authors:

Abstract and Figures

Recent studies attribute evolution of droplet size distribution in warm convective clouds to enhancement of collision-coalescence by turbulence. This study is aimed at better understanding of enhanced collisions and preferential concentration by analysis of droplet motion in vortex tubes: small coherent structures characteristic for high Reynolds number turbulent flows. Former research of such effects by Bajer et al., 2000 and by Hill, 2005 was limited to horizontally oriented vortex tubes only. Herein we analyze tubes approximated by point vortices of axes parallel or oblique to the direction of gravity. Droplet motion in a predefined line vortex undergoing stretching is governed by viscosity and gravity forces only, so Stokes equation to examine droplet trajectory is used. Both analytical and numerical solutions of Stokes equation allow to identify features such as stationary points, stationary orbits and limit cycles, which may influence preferential concentration and/or collisions. Stationary orbits exist in vortices aligned with gravity only and depend on all system parameters: droplet mass, vortex stretching strength and circulation. In contrast, such orbits do not exist in oblique vortices, but it is concluded from numerical simulations that under certain conditions droplets are attracted by limit cycles and/or by stationary points. Condition of existence of stationary point can be obtained analytically. For a given vortex circulation and stretching rate, stationary point can exist for the deflection from vertical exceeding certain critical value and for droplets exceeding certain critical radius. Stationary behavior is observed for two components of motion in a plane perpendicular to vortex axis only. Simulations of motion of numerous droplets of the same size, distributed homogeneously in space were conducted in an oblique vortex. For different sets of parameters these simulations showed appearance of stationary points, limit cycles or both. Figure shows droplets initiated in a a plane stable with respect to the motion along the axis (in such a case motion remains two-dimensional) and in unstable plane, where droplets move in third dimension along the vortex axis.
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= 2.5·104m2
s.
Values of other parameters used were as follows:
air kinematic viscosity µ= 1.776 ·105kg
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.
¨rr˙
φ2=(K1
2r+ ˙r+K3sin(φ))
2 ˙r˙
φ+r¨
φ=1
rr˙
φK3cos(φ))
¨z=K1z˙zK2
.(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=2
S.(5)
and on three in case with gravity direction nonpar-
allel to the vortex axis (θ6= 0):
K1=γτ =L1,
K2=2
Scosθ =L2cosθ,
K3=2
Ssinθ =L2sinθ.
(6)
L2parameter has a direct physical interpretation
as a rate of droplet terminal velocity ( ) 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) = z0z0b
λ+λ
[λ+exp(λt)λexp(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: z0bγ1R2cos 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 120µm 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±q1K1
K2
3
2
φst±=arcsin(1
2r1±q1K1
K2
3
2)
(9)
under the condition K2
3K1. This condition splits
into two:
L2
2L1,
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:
rstis always unstable,
rst+is unstable only under conditions: K1
(1
4,1
2)and K2
3K2
11
2K1+1
8
1
2K1.
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
... Such research for the simplest model of a line vortex with stretching was conducted by Markowicz et al. (2000) with limitation to horizontally oriented vortices. In order to better understand the problem of cloud droplet dynamics in atmospheric conditions, the same model but with arbitrary gravity alignment was studied by Karpinska and Malinowski (2014). Another model, free from the problem of unrealistic singularity on the vortex axis, is a Burgers vortex with stretching. ...
... What is more, every droplet has one unstable equilibrium point at z * 0 = S v A −1 cot θ = ν −1 gδτ p cos θ . The analytical solution was presented in Karpinska and Malinowski (2014). The solutions of Eq. (4) in 2-D space have several different attractors. ...
Article
Full-text available
The phenomenon of cloud voids, i.e., elongated volumes inside a cloud that are devoid of droplets, was observed with laser sheet photography in clouds at a mountain-top station. Two experimental cases, similar in turbulence conditions yet with diverse droplet size distributions and cloud void prevalence, are reported. A theoretical explanation is proposed based on the study of heavy inertial sedimenting particles inside a Burgers vortex. A general conclusion regarding void appearance is drawn from theoretical analysis. Numerical simulations of polydisperse droplet motion with realistic vortex parameters and Mie scattering visual effects accounted for can explain the presence of voids with sizes similar to that of the observed ones. Clustering and segregation effects in a vortex tube are discussed for reasonable cloud conditions.
... Such research for the simplest model of a line vortex with stretching was conducted by Markowicz et al. (2000) with limitation to horizontally oriented vortices. In order to better understand the problem of cloud droplet dynamics in atmospheric conditions the same model but with arbitrary gravity alignment was studied by Karpinska and Malinowski (2014). Another model, free from the problem of unrealistic singularity on the vortex axis, is a Burgers vortex with stretching. ...
... It is determined by stretching outflow drag and gravity. As a consequence, as found in (Karpinska and Malinowski, 2014), position z shows an exponential dependence on time. Every droplet has one unstable equilibrium point z * 0 = SvA −1 cot θ. ...
Preprint
Full-text available
The phenomenon of cloud voids, i.e., elongated volumes inside a cloud that are devoid of droplets, was observed with laser sheet photography in clouds at a mountain-top station. Two experimental cases, similar in turbulence conditions yet with diverse droplet size distributions and cloud void prevalence, are reported. A theoretical explanation is proposed based on the study of heavy inertial sedimenting particles inside a Burgers vortex. A general conclusion regarding void appearance is drawn from theoretical analysis. Numerical simulations of polydisperse droplet motion with realistic vortex parameters and Mie scattering visual effects accounted for can explain the presence of voids with sizes similar to that of the observed ones. Preferential concentration and sorting effects in a vortex tube are discussed for reasonable cloud conditions.
... Such research for the simplest model of a line vortex with stretching was conducted by Markowicz et al. (2000) with limitation to horizontally oriented vortices. In order to better understand the problem of cloud droplet dynamics in atmospheric conditions the same model but with arbitrary gravity alignment was studied by Karpinska and Malinowski (2014). Another model, free from the problem of unrealistic singularity on the vortex axis, is a Burgers vortex with stretching. ...
... It is determined by stretching outflow drag and gravity. As a consequence, as found in (Karpinska and Malinowski, 2014), position z shows an exponential dependence on time. Every droplet has one unstable equilibrium point z * 0 = SvA −1 cot θ. ...
Preprint
Full-text available
The phenomenon of "cloud voids", i.e., elongated volumes inside a cloud that are devoid of droplets, was observed with laser sheet photography in clouds at a mountain-top station. Two experimental cases, similar in turbulence conditions yet with diverse droplet size distributions and cloud void prevalence, are reported. A theoretical explanation is proposed based on the study of heavy inertial sedimenting particles inside a Burgers vortex. A general conclusion regarding void appearance is drawn from theoretical analysis. Numerical simulations of polydisperse droplet motion with realistic vortex parameters and Mie scattering visual effects accounted for can explain the presence of voids with sizes similar to that of the observed ones. Preferential concentration and sorting effects in a vortex tube are discussed for reasonable cloud conditions.
Conference Paper
Full-text available
The simple adiabatic parcel model of the condensational growth of cloud droplets fails to explain the creation of droplets large enough to initiate rain on the observed time-scale. Shaw et al. (1998, see also Grabowski & Vaillancourt 1999) put forward a mechanism of accelerated condensation induced by small-scale non-uniformities in the concentration of droplets. A pocket of lower concentration would have higher level of supersaturation and therefore droplets in such a pocket should grow faster. In their model local concentration deficits are caused by the small-scale vortices, believed to form spontaneously at the dissipative scales of developed turbulence. They considered the purely two-dimensional process of centrifugal expulsion from the vicinity of a vortex patch of droplets dragged around by the Stokes's force. Here we consider a different effect related to the small-scale vortical structures in turbulence. There is still much uncertainty about the mechanisms of the formation of such structures but all candidates considered involve vortex stretching. The individual filaments can emerge due to the concentration and enhancment of the background vorticity or they can form when a stretched vortex sheet either breaks up or rolls up from its ends (Jimenez 1999). We show that the combined effects of gravity and of the irrotational component of the flow responsible for vortex stretching are more relevant for the "growth gap" problem for two reasons. Firstly, they accelerate condensational growth by creating pockets of concentration deficit on a faster time-scale than the purely centrifugal evacuation and, secondly, they induce high relative velocities of the droplets thus signifficantly increasing the probability of the formation of large droplets by coalescense.
Article
Full-text available
This paper presents a linear stability analysis as well as some numerical results for the motion of heavy particles in the flow field of a Burgers vortex, under the combined effects of particle inertia, Stokes drag, and gravity. By rendering the particle motion equations dimensionless, the particle Stokes number, a Froude number, and a vortex Reynolds number are obtained as the governing three parameters. In the absence of gravity, the vortex center represents a stable equilibrium point for particles up to a critical value of the Stokes number, as the inward drag overcomes the destabilizing centrifugal force on the particle. Particles exceeding the critical Stokes number value asymptotically approach closed circular orbits. Under the influence of gravity, one or three equilibrium points appear away from the vortex center. Both their locations and their stability characteristics are derived analytically. These stability characteristics can furthermore be related to the nature of the critical points in a related directional force field. These findings are expected to be applicable to the coupling between the small‐scale turbulent flow structures and the motion of suspended particles.
Article
Full-text available
In this survey we consider the impact of turbulence on cloud formation from the cloud scale to the droplet scale. We assess progress in understanding the effect of turbulence on the condensational and collisional growth of droplets and the effect of entrainment and mixing on the droplet spectrum. The increasing power of computers and better experimental and observational techniques allow for a much more detailed study of these processes than was hitherto possible. However, much of the research necessarily remains idealized and we argue that it is those studies which include such fundamental characteristics of clouds as droplet sedimentation and latent heating that are most relevant to clouds. Nevertheless, the large body of research over the last decade is beginning to allow tentative conclusions to be made. For example, it is unlikely that small-scale turbulent eddies (i.e. not the energy-containing eddies) alone are responsible for broadening the droplet size spectrum during the initial stage of droplet growth due to condensation. It is likely, though, that small-scale turbulence plays a significant role in the growth of droplets through collisions and coalescence. Moreover, it has been possible through detailed numerical simulations to assess the relative importance of different processes to the turbulent collision kernel and how this varies in the parameter space that is important to clouds. The focus of research on the role of turbulence in condensational and collisional growth has tended to ignore the effect of entrainment and mixing and it is arguable that they play at least as important a role in the evolution of the droplet spectrum. We consider the role of turbulence in the mixing of dry and cloudy air, methods of quantifying this mixing and the effect that it has on the droplet spectrum. Copyright © 2012 Royal Meteorological Society and British Crown Copyright, the Met Office
Article
Full-text available
The effect of turbulence on the collision rate between droplets in clouds is investigated. Because of their inertia, water droplets can be shot out of curved streamlines of the turbulent airflow. The contribution of such a "sling effect" in the collision rate of the same-size water droplets is described and evaluated. It is shown that already for turbulence with the dissipation rate 103 cm2 sL-3, the sling effect gives a contribution to the collision rate of 15-μm droplets comparable to that due to the local velocity gradient. That may explain why the formulas based on the local velocity gradient consistently underestimate the turbulent collision rate, even with the account of preferential concentration.
Article
Droplet velocities, concentrations, and geometric collision rates are calculated for droplets falling into Burgers vortices as a step toward understanding the role of turbulence-induced collisions of cloud water droplets. The Burgers vortex is an often used model of vortices in high Reynolds number turbulence. Droplet radii considered are 10, 20, and 40 mum those radii are relevant to warm rain initiation. A method of calculating the concentrations of droplets along their trajectories by means of differential geometry is derived and implemented. A generalization of the rate of geometrical collisions of inertial particles is derived; the formulation applies for any local vorticity and rate of strain, and the classic collision-rate formula is obtained in the process. The relative velocities of droplets of different radii and their spatial variation of concentration affects spatial variation of collision rate; greater variation exists for a stronger vortex. The physical effects included in the droplet equation of motion are inertia, gravity, viscous drag, pressure and shear stress, added mass, the history integral, and the lift force. The lift force requires calculation of droplet angular velocity, the equation for which contains rotational inertial and viscous drag. An initial condition is found that does not cause an impulse in the history integral. The important terms in the droplets' equations of motion are found such that simpler approximate equations can be used. It is found that the lift force is negligible, the history integral is not. For smaller droplets in regions of lower vorticity, the time derivative of the difference of slip velocity and gravitationally induced drift velocity may be neglected. The present study suggests that acceleration-induced coalescence is most significant for droplets that are entrained into or formed within an intensifying vortex as distinct from falling toward the vortex.
Article
Turbulence is ubiquitous in atmospheric clouds, which have enormous turbulence Reynolds numbers owing to the large range of spatial scales present. In-deed, the ratio of energy-containing and dissipative length scales is on the order of 10 5 for a typical convective cloud, with a corresponding large-eddy Reynolds number on the order of 10 6 to 10 7 . A characteristic trait of high-Reynolds-number turbulence is strong intermittency in energy dissipation, Lagrangian acceleration, and scalar gradi-ents at small scales. Microscale properties of clouds are determined to a great extent by thermodynamic and fluid-mechanical interactions between droplets and the surround-ing air, all of which take place at small spatial scales. Furthermore, these microscale properties of clouds affect the efficiency with which clouds produce rain as well as the nature of their interaction with atmospheric radiation and chemical species. It is expected, therefore, that fine-scale turbulence is of direct importance to the evolution of, for example, the droplet size distribution in a cloud. In general, there are two levels of interaction that are considered in this review: (a) the growth of cloud droplets by condensation and (b) the growth of large drops through the collision and coalescence of cloud droplets. Recent research suggests that the influence of fine-scale turbulence on the condensation process may be limited, although several possible mechanisms have not been studied in detail in the laboratory or the field. There is a growing con-sensus, however, that the collision rate and collision efficiency of cloud droplets can be increased by turbulence-particle interactions. Adding strength to this notion is the growing experimental evidence for droplet clustering at centimeter scales and below, most likely due to strong fluid accelerations in turbulent clouds. Both types of interac-tion, condensation and collision-coalescence, remain open areas of research with many possible implications for the physics of atmospheric clouds.
Review article. droplet growth in warm turbulent clouds
  • B J Devenish
Devenish, B. J., and Coauthors, 2012: Review article. droplet growth in warm turbulent clouds. Q. J. R. Meteorol. Soc., 138, 1401–1429, doi:10.1002/qj. 1897.