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A Lagrangian approach to droplet condensation in turbulent clouds Rutger IJzermans, Michael W. Reeks...
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A Lagrangian approach to droplet condensation in turbulent clouds
Rutger IJzermans, Michael W. ReeksSchool of Mechanical & Systems EngineeringNewcastle University, United Kingdom
Ryan SidinDepartment of Mechanical EngineeringUniversity of Twente, the Netherlands
Objective and motivation
Research question: How does turbulence influence
condensational growth of droplets?
Application: Rain initiation in atmospheric clouds
Objectives: - Gain understanding of rain initiation process, from cloud condensation
nuclei to rain droplets
- Elucidate role of turbulent macro-scales and micro-
scales on condensation of droplets in clouds
Background: scales in turbulent clouds
Turbulence:Large scales: L0 ~ 100 m, 0 ~ 103 s, u0 ~ 1 m/s,Small scales: ~ 1 mm, k ~ 0.04 s, uk ~ 0.025 m/s.
Droplets: Radius: Inertia: Settling velocity:Formation: rd ~ 10-7 m, St = d/k ~ 2 × 10-6, vT/uk ~ 3 × 10-5
Microscales: rd ~ 10-5 m, St = d/k ~ 0.02, vT/uk ~ 0.3
Rain drops: rd ~ 10-3 m, St = d/k ~ 200, vT/uk ~ 3000
COLLISIONS / COALESCENCE
CONDENSATION
Collisions / coalescence process vastly enhancedif droplet size distribution at micro-scales is broad
Classic theory (Twomey (1959); Shaw (2003)):
Fluid parcel, filled with many droplets of different sizes
Droplet size distribution at microscales
If parcel rises, temperature decreases due toadiabatic expansion, and supersaturation s increases:
Problem:• Droplet size distribution in reality (experiments) becomes broader
O()Droplet growth is given by:
or: Size distribution PDF(rd)becomes narrower in time!
Twomey’s fluid parcel approximation is not allowed in turbulence
Cloud turbulence modelled by kinematic simulation:
• All relevant flow scales can be incorporated by choosing kn of appropriate length• Turbulent energy spectrum required as input
Numerical model for condensation in cloud
Ideally, Direct Numerical Simulation of:• Velocity and pressure fields (Navier-Stokes)• Supersaturation and temperature fieldsComputationally too expensive: O(L0/)3 ~ 1015 cells- State-of-the-art DNS: 5123 modes, L0 ~ 70cm (Lanotte et al., J. Atm. Sci. (2008))
Full condensation model
- rate-of-change of droplet mass md:
- rate-of-change of mixture temperature T:
- rate-of-change of supersaturation s:
Droplet modelled as passive tracer, contained within a moving air parcel:
Along its trajectory (Lagrangian):
Mixture of air & water vapour
Latent heat release
Adiabatic cooling
Vapour depletion
volume Vp
Simplified condensation model
Track droplets as passive tracers:
Rate-of-change of droplet mass md:
Temperature T and supersaturation s are assumed to depend on adiabatic cooling only:
Air & water vapor
Typical supersaturation profile
Imposed mean temperature and supersaturation profiles:
1340 1360 1380279
280
-0.01
0
0.01
0.02
height w.r.t. earth' s surface: z (m)
tem
pera
ture
:T(K
)
supe
rsat
urat
ion:
s
500 1000 1500 2000270
275
280
285
290
295
300
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
sT
Focus on regions where supersaturation is close to zero
Determine droplet size distribution:
• Droplet population (Nd=8000) initially randomly distributed in a plane at time t = te and height ze:
• Droplet trajectories traced backward in time: t = te 0
• At t = 0 a monodisperse distribution is assumed: rd (0) = r0 = 10-7m
• Condensation model equations are integrated forward in time to obtain droplet size distribution in the plane at t = te
Computational strategy
Results: dispersion in 3D KS-flow field
1-particle statistics: Short times: <|x – x0|2> ~ t2
Long times: <|x – x0|2> ~ t
2-particle statistics:
time (0)
<|x-
x 0|2 >(k
0-2)
10-2 10-1 100 101 102 103 104 105 106100
102
104
106
108
1010
1012
2
1
time (0)
<(r/
r 0)2 >
100 101 102 103 104 105 10610-1
101
103
105
107
109
1011
6
3
1
Slope = 4.5, similar to [Thomson & Devenish, J.F.M. 2005]
time (s)
vert
ical
posi
tion:
z(m
)
0 50 100 150 2001450
1500
1550
1600
1650
r0 =
Flight of 2 particles initially separated by distance r0=:
In agreement with Taylor (1921)
Time evolution of droplet position and size
ze = 1355 m ; te =100 s ; size of sampling area = 1 x 1 cm2
Backward tracing: t = te 0 Forward tracing: t = 0 te
Droplet evaporation in regions where s < 0
time (s)
Nd/
Nd,
max
20 40 60 80 1000.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1z0 = 1340 mz0 = 1350 mz0 = 1360 mz0 = 1370 mz0 = 1380 m
Number of droplets at various altitudes:
Rapid initial evaporation
Forward tracing: t = 0 tc
time (s)
0
20
40
60
80
100r (m)0
1E-052E-05
3E-054E-05
f(r,
t)
0
100000
200000
300000
400000
Droplet radius distributions in time
Temporal evolution of radius distribution function (z0 = 1355 m):
droplet radius: r (m)
radi
usdi
stri
butio
n:f
(m-1
)
radi
usdi
stri
butio
n:f
(m-1
)
10-7 10-6 10-5
2E+07
4E+07
6E+07
8E+07
1E+08
1.2E+08
2E+06
4E+06
6E+06
8E+06
1E+07
t = 0.1 st = 0.5 st = 1.0 st = 5.0 st = 10 st = 20 s
time (s)
0
20
40
60
80
100r (m)
01E-05
2E-053E-05
4E-05
f(r ,
t)
0
200000
400000
Radius distributions after te = 100 s
Influence of measurement altitude:(size of sampling area L = 500 m)
Influence of sampling area width: (ze = 1350 m)
droplet radius: r (m)
radi
usdi
stri
butio
nfu
nctio
n:f(
r)(m
-1)
0 2E-05 4E-050
50000
100000
150000
200000L = 500 mL = 100 mL = 10 mL = 1 mL = 0.1 mL = 0.01 m
droplet radius: r (m)
radi
usdi
stri
butio
nfu
nctio
n:f
(m-1
)
0 1E-05 2E-05 3E-05 4E-05 5E-05 6E-050
20000
40000
60000
80000
z = 1340 mz = 1350 mz = 1360 mz = 1370 mz = 1380 m
Effect of different scales in turbulence
Droplet radius distribution in flow with:- Only large scales included (n=1-10)- Only small scales included(n=191-200)- Wide range of scales included(n=1-200)
ze = 1350m, te = 100s, L = 0.01m
Results for two-way coupled model
Eulerian evolution of droplet size distribution for nd = (5 η)-3 = 8.0 x 106 m-3:
ze = 1350 m ze = 1380m
Results two-way coupled model: interpretation
Saturation of droplet radius distribution functionfollows from a balance between: - Adiabatic expansion (“forcing”)- Vapour depletion (“damping” with time scale s)- Latent heat release (“damping” with time scale L)
Equation for supersaturation s is:
with:
This can be rewritten into:
Results two-way coupled model: interpretation
Relative importance ofthe two damping terms, s/L,as a funciton of temperature:
Dependence of vapour depletiontime scale s on droplet radius rd
and on droplet number density nd:
Results two-way coupled model
te=100s,ze = 1350m,nd=(2)-3 = 0.125 x 109 m-3
Influence of length of the sampling area L:
Conclusions
• Droplet size distribution may become broader during condensation:
- Large scales of turbulent motion responsible for transport of dropletsto different regions of the flow, with different supersaturations
- Small scales of turbulent motion responsible forlocal mixing of large and small droplets
• Broad droplet size distribution observed both in simplifiedcondensation model and in two-way coupled condensation model
• Broadening of droplet size distribution enhanced by:- Higher flow velocities (more vigourous turbulence)- Lower droplet number density- Lower surrounding temperatures