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))

Energy spectrum in wavenumber space

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,L=500m

Influence of 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