P a rt icle Colli s ion s in Tur b u len t Flow s Michel Vok u hle - - PowerPoint PPT Presentation

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Particle Collisions in Turbulent Flows

Michel Voßkuhle

Laboratoire de Physique

13 December 2013

ÉCOLE NORMALE SUPÉRIEURE DE LYON

• M. Voßkuhle › Particle Collisions in Turbulent Flows

2

ÉCOLE NORMALE SUPÉRIEURE DE LYON

• M. Voßkuhle › Particle Collisions in Turbulent Flows

2

ÉCOLE NORMALE SUPÉRIEURE DE LYON

• M. Voßkuhle › Particle Collisions in Turbulent Flows

2

ÉCOLE NORMALE SUPÉRIEURE DE LYON

• M. Voßkuhle › Particle Collisions in Turbulent Flows

2

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Particle Collisions in Turbulent Flows

Outline

Introduction Prevalence of sling/caustics/RUM effect Multiple collisions KS vs. DNS

• M. Voßkuhle › Particle Collisions in Turbulent Flows

3

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Rain formation and the droplet size distribution

1 10 102 103 104 drop radius [µm] relative mass 10 20 30 time [min]

Lamb Meteorol. Monogr. (2001);Shaw ARFM (2003)

∂ f (a) ∂t = nucleation/ condensation + collision/ coalescence

• M. Voßkuhle › Particle Collisions in Turbulent Flows

4

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Rain formation and the droplet size distribution

1 10 102 103 104 drop radius [µm] relative mass 10 20 30 time [min]

Lamb Meteorol. Monogr. (2001);Shaw ARFM (2003)

∂ f (a) ∂t = nucleation/ condensation + 1 2 ∫

a

a2 a′′2 Γ(a′′, a′)f (a′′)f (a′)da′ − ∫

∞

Γ(a, a′)f (a)f (a′)da′

a′′3 = a3 − a′3

• M. Voßkuhle › Particle Collisions in Turbulent Flows

4

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Rain formation and the droplet size distribution

1 10 102 103 104 drop radius [µm] relative mass 10 20 30 time [min]

Lamb Meteorol. Monogr. (2001);Shaw ARFM (2003)

∂ f (a) ∂t = nucleation/ condensation + 1 2 ∫

a

a2 a′′2 Γ(a′′, a′)f (a′′)f (a′)da′ − ∫

∞

Γ(a, a′)f (a)f (a′)da′

a′′3 = a3 − a′3

Many Other Applications › planet growth in

protoplanetary disks

› diesel sprays › ...

• M. Voßkuhle › Particle Collisions in Turbulent Flows

4

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Introductory example:

Kinetic theory

Collision cylinder

a 2a ⟨w⟩ ∆t

› simplification:

• nly one particle size

› collision rate for one particle

Rc = n π(2a)2⟨w⟩

› overall collision rate

N c = 1 2n2 π(2a)2⟨w⟩

• Γkin(a)

Collision kernel

Γkin(a) = π(2a)2⟨w⟩

• M. Voßkuhle › Particle Collisions in Turbulent Flows

5

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

(Inertial) Particle collisions in Turbulent Flows

› finite density

ρp > ρf

› finite size

0 < a ≪ η

› equations of motion:

dX dt = V, dV dt = u(X, t) − V τp +G

Maxey & Riley Phys. Fluids (1983) Gatignol J. méc. théor. appl. (1983)

› dimensionless quantity:

Stokes number St = τp τK = 2 9 ρp ρf a2 η2

DNS

› Navier–Stokes equations › periodic box › 3843 grid points › Reλ = 130

Kinematic Simulations

› synthetic turbulence › efficient

Fung et al. JFM (1992)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

6

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Determining the collision rate

› part of simulation box

with particles

• M. Voßkuhle › Particle Collisions in Turbulent Flows

7

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Determining the collision rate

› part of simulation box

with particles

› divide into segments › know which particles in

which cell

• M. Voßkuhle › Particle Collisions in Turbulent Flows

7

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Determining the collision rate

› part of simulation box

with particles

› divide into segments › know which particles in

which cell

› consider only

surrounding cells

• M. Voßkuhle › Particle Collisions in Turbulent Flows

7

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Determining the collision rate

extrapolation is inexact rather use interpolation

Collision kernel

Nc(T) T Vsys = N c = 1 2n2Γ(a)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

8

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Prevalence of the sling/caustics/RUM effect

• M. Voßkuhle › Particle Collisions in Turbulent Flows

9

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Saffman & Turner JFM (1956)

› St → 0: particles follow flow

Rc = n ∫ −wr(2a, Ω)Θ[−wr(2a, Ω)] dΩ

› average to obtain total collision rate

N c = 1 2 n

2 ∫ 1

2⟨∣wr(2a)∣⟩ dΩ

› approximate ⟨∣wx(2a)∣⟩ = 2a ⟨∣∂ux/∂x∣⟩ › assume Gaussian statistics with

⟨(∂ux/∂x)2⟩ = ε/15ν

2a a ΓST = (8π 15 )

1/2 (2a)3

τK

• M. Voßkuhle › Particle Collisions in Turbulent Flows

10

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision kernel(s)

St → 0 Saffman & Turner JFM (1956)

ΓST = (8π 15 )

1/2 (2a)3

τK

St → ∞ Abrahamson Chem. Eng. Sci. (1975)

ΓA = Γkin with Vrms = (η/τK)f(St, Reλ)

ΓA = 4 √ π(2a)2 η τK f(St, Reλ)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

11

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision kernel(s)

St → 0 Saffman & Turner JFM (1956)

ΓST = (8π 15 )

1/2 (2a)3

τK Preferential concentration ?

20 40 60 80 100 1 2 3 4 5 ΓτK/(2a)3 St

Sling/caustics/RUM effect ?

St → ∞ Abrahamson Chem. Eng. Sci. (1975)

ΓA = Γkin with Vrms = (η/τK)f(St, Reλ)

ΓA = 4 √ π(2a)2 η τK f(St, Reλ)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

11

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Preferential concentration

› mathematically exact ΓSC = 2π(2a)2g(2a)⟨∣wr∣⟩ › radial distribution function g(r)

g(r) ∼ (r/η)

(D2−3),

r/η ≪ 1

Sundaram & Collins JFM (1997)

100 200 300 400 400 500 600 700 x (pixels) y (pixels) 10 20 30 40 50 40 50 60 70 x (mm) y (mm)

Monchaux et al. Phys. Fluids (2010)

5 10 15 20 25 30 35 1 2 3 4 5 6 g(2a) St Reλ = 130

• M. Voßkuhle › Particle Collisions in Turbulent Flows

12

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Sling/caustics/RUM effect

Sling

› particles slung by

vortices

Falkovich et al. Nature (2002)

Caustics

X V t = t0 t > t0

› faster particles

“overtake” slower ones

› points in phase space

multi-valued

Wilkinson, Mehlig, & Bezuglyy PRL (2006)

RUM

Random Uncorrelated Motion

› two contributions to

motion of inertial particles » smooth spatially correlated movement » random uncorrelated movement

Simonin et al. Phys. Fluids (2006) Reeks et al. Proc. FEDSM (2006)

Γ = ΓST

• ∼a3

g(2a)

• ∼aD2−3

+ ΓAhS(St, Reλ)

• ∼a2
• M. Voßkuhle › Particle Collisions in Turbulent Flows

13

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

St = 2 9 ρp ρf a2 η2

△ ρp/ρf = 250 ⇒ a = 2a0 □ ρp/ρf = 1000 ⇒ a = a0

• ρp/ρf = 4000

⇒ a = 1 2a0

• M. Voßkuhle › Particle Collisions in Turbulent Flows

14

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

St = 2 9 ρp ρf a2 η2

△ ρp/ρf = 250 ⇒ a = 2a0 □ ρp/ρf = 1000 ⇒ a = a0

• ρp/ρf = 4000

⇒ a = 1 2a0

Saffman & Turner

5 10 15 20 25 30 35 40 0.1 0.2 0.3 0.4 0.5 ΓτK/(2a)3 St

Sling/caustics/RUM

50 100 150 200 250 300 350 1 2 3 4 5 6 ΓτK/ [η(2a)2] St

Voßkuhle et al. arXiv: 1307.6853 [physics.flu-dyn]

• M. Voßkuhle › Particle Collisions in Turbulent Flows

14

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Γ = ΓST

• ∼a3

g(2a)

• ∼aD2−3

+ ΓAhS(St, Reλ)

• ∼a2

› Expected values for

Γ(a0)/Γ(2a0) = Γ(a0/2)/Γ(a0) ∼ a

2

⇒ Γ(a0)/Γ(2a0) = 1 4 ∼ a

3

⇒ Γ(a0)/Γ(2a0) = 1 8 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 Γ(a0)/Γ(2a0) St

Voßkuhle et al. arXiv: 1307.6853 [physics.flu-dyn]

• M. Voßkuhle › Particle Collisions in Turbulent Flows

15

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Γ = ΓST

• ∼a3

g(2a)

• ∼aD2−3

+ ΓAhS(St, Reλ)

• ∼a2

› Expected values for

Γ(a0)/Γ(2a0) = Γ(a0/2)/Γ(a0) ∼ a

2

⇒ Γ(a0)/Γ(2a0) = 1 4 ∼ a

3

⇒ Γ(a0)/Γ(2a0) = 1 8 ∼ a

3a D2−3

⇒ Γ(a0)/Γ(2a0) = (1 2)

D2

0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 Γ(a0)/Γ(2a0) St

Voßkuhle et al. arXiv: 1307.6853 [physics.flu-dyn]

• M. Voßkuhle › Particle Collisions in Turbulent Flows

15

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision velocities

› Cumulative PDF of relative velocity F(∣wr∣) = ∫

∣wr∣

p(∣wr∣)dwr

“How many particle pairs have relative velocities smaller than ∣wr∣?”

› Cumulative distribution of the flux φ(∣wr∣) = ∫

∣wr∣

∣wr∣ ⟨∣wr∣⟩p(∣wr∣)dwr

“How many colliding particles have relative velocities smaller than ∣wr∣?”

0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 2 4 6 8 10 ∣wr∣/uK ∣wr∣ τK/(2a) St = 0.2

• M. Voßkuhle › Particle Collisions in Turbulent Flows

16

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision velocities

› Cumulative PDF of relative velocity F(∣wr∣) = ∫

∣wr∣

p(∣wr∣)dwr

“How many particle pairs have relative velocities smaller than ∣wr∣?”

› Cumulative distribution of the flux φ(∣wr∣) = ∫

∣wr∣

∣wr∣ ⟨∣wr∣⟩p(∣wr∣)dwr

“How many colliding particles have relative velocities smaller than ∣wr∣?”

0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.20.30.40.50.60.7 2 4 6 8 10 ∣wr∣/uK ∣wr∣ τK/(2a) St = 0.3

• M. Voßkuhle › Particle Collisions in Turbulent Flows

16

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision velocities

› Cumulative PDF of relative velocity F(∣wr∣) = ∫

∣wr∣

p(∣wr∣)dwr

“How many particle pairs have relative velocities smaller than ∣wr∣?”

› Cumulative distribution of the flux φ(∣wr∣) = ∫

∣wr∣

∣wr∣ ⟨∣wr∣⟩p(∣wr∣)dwr

“How many colliding particles have relative velocities smaller than ∣wr∣?”

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 2 4 6 8 10 ∣wr∣/uK ∣wr∣ τK/(2a) St = 0.75

• M. Voßkuhle › Particle Collisions in Turbulent Flows

16

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision velocities

› Cumulative PDF of relative velocity F(∣wr∣) = ∫

∣wr∣

p(∣wr∣)dwr

“How many particle pairs have relative velocities smaller than ∣wr∣?”

› Cumulative distribution of the flux φ(∣wr∣) = ∫

∣wr∣

∣wr∣ ⟨∣wr∣⟩p(∣wr∣)dwr

“How many colliding particles have relative velocities smaller than ∣wr∣?”

0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 2 4 6 8 10 ∣wr∣/uK ∣wr∣ τK/(2a) St = 1.0

• M. Voßkuhle › Particle Collisions in Turbulent Flows

16

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision velocities

› Cumulative PDF of relative velocity F(∣wr∣) = ∫

∣wr∣

p(∣wr∣)dwr

“How many particle pairs have relative velocities smaller than ∣wr∣?”

› Cumulative distribution of the flux φ(∣wr∣) = ∫

∣wr∣

∣wr∣ ⟨∣wr∣⟩p(∣wr∣)dwr

“How many colliding particles have relative velocities smaller than ∣wr∣?”

0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 2 4 6 8 10 ∣wr∣/uK ∣wr∣ τK/(2a) St = 2.5

• M. Voßkuhle › Particle Collisions in Turbulent Flows

16

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Collision velocities

› Cumulative PDF of relative velocity F(∣wr∣) = ∫

∣wr∣

p(∣wr∣)dwr

“How many particle pairs have relative velocities smaller than ∣wr∣?”

› Cumulative distribution of the flux φ(∣wr∣) = ∫

∣wr∣

∣wr∣ ⟨∣wr∣⟩p(∣wr∣)dwr

“How many colliding particles have relative velocities smaller than ∣wr∣?”

0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 2 4 6 8 10 ∣wr∣/uK ∣wr∣ τK/(2a) St = 5.0

• M. Voßkuhle › Particle Collisions in Turbulent Flows

16

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Multiple collisions

• M. Voßkuhle › Particle Collisions in Turbulent Flows

17

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Ghost collisions and the Saffman–Turner theory

Saffman & Turner JFM (1956)

› flow is locally hyperbolic › flow is persistent › fluid element can pass

through collision sphere repeatedly ⇒ spurious “ghost” collisions

Brunk et al. JFM (1998) Andersson et al. EPL (2007) Gustavsson, Mehlig, & Wilkinson NJP (2008)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

18

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Ghost collisions and the Saffman–Turner theory

Saffman & Turner JFM (1956)

⇒

Particles kept in flow may collide again

Brunk et al. JFM (1998) Andersson et al. EPL (2007) Gustavsson, Mehlig, & Wilkinson NJP (2008)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

18

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Overestimation by Ghost collisions

0.05 0.1 0.15 0.2 1 2 3 4 5 Γm/Γ1 St 20 40 60 80 100 ΓτK/(2a)3

Voßkuhle et al. PRE (2013)

ΓGCA ∶ ghost collision approximation Γ1 ∶ only first collisions Γm ∶ multiple collisions ΓGCA = Γ1 + Γm

› Ghost collisions overestimate

collision kernel by up to 20 %

› Relative estimation error falls

with Stokes number

Consistent with previous results by Brunk et al. JFM (1998) and Wang et al. Phys. Fluids (1998)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

19

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Statistics of multiple collisions

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 2 4 6 8 10 P(Nc∣Nc ≥ 1) Nc St = 0.5 1.0 2.0 4.0

Voßkuhle et al. PRE (2013)

› Multiple collisions statistics

follow P(Nc∣Nc ≤ 1) = β(St)α(St)Nc

› Markovian interpretation:

particle has probability α(St) to collide again

› Simple model

Γm = Γ1

∞

∑

Nc=2

βαNc = Γ1 βα2 1 − α

• M. Voßkuhle › Particle Collisions in Turbulent Flows

20

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Statistics of multiple collisions

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 2 4 6 8 10 P(Nc∣Nc ≥ 1) Nc St = 0.5 1.0 2.0 4.0

Voßkuhle et al. PRE (2013)

› Multiple collisions statistics

follow P(Nc∣Nc ≤ 1) = β(St)α(St)Nc

› Markovian interpretation:

particle has probability α(St) to collide again

› Simple model

Γm = Γ1

∞

∑

Nc=2

βαNc = Γ1 βα2 1 − α

1 2 3 4 5 6 1 2 3 4 5 ΓmτK/(2a)3 St model

DNS

• M. Voßkuhle › Particle Collisions in Turbulent Flows

20

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Distance between trajectories

0.2 0.4 0.6 0.8 0 5 10 15 202530354045 distance/2π t/TL St = 1.0

d(t2) d(t1) 2 4 6 8 10 10 12 14 16 18 20 distance/2a t/TL

• M. Voßkuhle › Particle Collisions in Turbulent Flows

21

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Pair separation

Jullien et al. PRL (1999)

› results for tracers › starting with similar initial conditions

Scatamacchia et al. PRL (2012) Rast & Pinton PRL (2011)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

22

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Pair separation

Jullien et al. PRL (1999)

› results for tracers › starting with similar initial conditions

Scatamacchia et al. PRL (2012) Rast & Pinton PRL (2011)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

22

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Contact time distribution

dc distance time ∆t1 ∆t2 ⋯

• M. Voßkuhle › Particle Collisions in Turbulent Flows

23

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Contact time distribution

dc distance time ∆t1 ∆t2 ⋯

› exponential tail for long time › power law for short time › independent of distance dc

P(∆t1) for St = 1.5

10−6 10−4 10−2 100 102 0 1 2 3 4 5 6 7 8 9 10 ∆t1/TL

Voßkuhle et al. PRE (2013)

10−6 10−4 10−2 100 102 10−3 10−2 10−1 100 101 ∆t1/TL dc = 4a dc = 2a dc = a dc = a/2

• M. Voßkuhle › Particle Collisions in Turbulent Flows

24

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Contact time distribution

dc distance time ∆t1 ∆t2 ⋯

› exponential tail persists › power law vanishes › independent of collision count

P(∆ti), i > 1 for St = 1.5

10−5 10−4 10−3 10−2 10−1 100 101 0 1 2 3 4 5 6 7 8 9 10 ∆ti/TL

Voßkuhle et al. PRE (2013)

10−5 10−4 10−3 10−2 10−1 100 101 10−3 10−2 10−1 100 101 ∆ti/TL ∆t4 ∆t3 ∆t2 ∆t1∣Nc>1

• M. Voßkuhle › Particle Collisions in Turbulent Flows

25

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Contact time distribution

dc distance time ∆t1 ∆t2 ⋯

› exponential tail persists › power law vanishes › independent of collision count

P(∆ti), i > 1 for St = 1.5

10−5 10−4 10−3 10−2 10−1 100 101 0 1 2 3 4 5 6 7 8 9 10 ∆ti/TL

Voßkuhle et al. PRE (2013)

10−5 10−4 10−3 10−2 10−1 100 101 10−3 10−2 10−1 100 101 ∆ti/TL ∆t4 ∆t3 ∆t2 ∆t1∣Nc>1

Kinetic theory

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 10−1 100 101 (2a/σ) P(∆t σ/[2a]) ∆t σ/(2a)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

25

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Contact time distribution

dc distance time ∆t1 ∆t2 ⋯

› exponential tail persists › power law vanishes › independent of collision count

P(∆ti), i > 1 for St = 1.5

10−5 10−4 10−3 10−2 10−1 100 101 0 1 2 3 4 5 6 7 8 9 10 ∆ti/TL

Voßkuhle et al. PRE (2013)

10−5 10−4 10−3 10−2 10−1 100 101 10−3 10−2 10−1 100 101 ∆ti/TL ∆t4 ∆t3 ∆t2 ∆t1∣Nc>1

• M. Voßkuhle › Particle Collisions in Turbulent Flows

25

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Contact time distribution

Different St

10−8 10−6 10−4 10−2 100 2 4 6 8 10 12 14 ∆t1/TL St = 4.0 St = 2.0 St = 1.0 St = 0.5

Voßkuhle et al. PRE (2013)

P(∆T1) ∼ e−κ∆t1/TL ∆t1 TL

−ξ

0.5 1 1.5 2 2.5 1 2 3 4 5 St κ ξ

• M. Voßkuhle › Particle Collisions in Turbulent Flows

26

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Multiple collisions and sling/caustics/RUM effect

Collision velocities

1 2 3 4 5 1 2 3 4 5 ⟨∣wr∣⟩c,i/uK St i = 1 i = m

Voßkuhle et al. PRE (2013)

› multiple collisions

have small relative velocities

› multiple collisions

stem from continuous collisions

› sling/caustics/RUM

effect does not lead to multiple collisions

• M. Voßkuhle › Particle Collisions in Turbulent Flows

27

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Kinematic Simulations vs. Direct Numerical Simulations

• M. Voßkuhle › Particle Collisions in Turbulent Flows

28

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Kinematic Simulations

u(x, t) =

Nk

∑

n=1

An cos(kn ⋅ x + ωnt) + Bn sin(kn ⋅ x + ωnt) An ⋅ kn = Bn ⋅ kn = 0 A2

n = B2 n = E(kn)∆kn

E(kn) ∼ k−5/3

n

› efficient › highly “turbulent” flows › widely used › “toy model”

Fung et al. JFM (1992)

• M. Voßkuhle › Particle Collisions in Turbulent Flows

29

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

Qualitatively the same for KS...

1 2 3 4 5 1 2 3 4 5 ΓτK/(2a)3 St ΓGCA Γ1 Γm

Voßkuhle et al. J. Phys.: Conf. Ser. (2011)

0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 Γm/Γ1 St 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 2 4 6 8 10 12 P(Nc∣Nc ≥ 1) Nc 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 0 1 2 3 4 5 6 7 8 9 TL P(∆t/TL) ∆t/TL St = 0.5 1.0 2.0 4.0

• M. Voßkuhle › Particle Collisions in Turbulent Flows

30

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

...but quantitatively very different from DNS

20 40 60 80 100 1 2 3 4 5 ΓGCAτK/(2a)3 St

KS DNS

Voßkuhle et al. J. Phys.: Conf. Ser. (2011)

0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 Γm/Γ1 St

KS DNS

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 2 4 6 8 10 12 P(Nc∣Nc ≥ 1) Nc 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 0 1 2 3 4 5 6 7 8 9 TL P(∆t/TL) ∆t/TL St = 0.5 1.0 2.0 4.0

• M. Voßkuhle › Particle Collisions in Turbulent Flows

30

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

0.2 0.4 0.6 0.8 1.0 ωmax 4.1 8.2 12.3 16.4 20.5 ωrms

• M. Voßkuhle › Particle Collisions in Turbulent Flows

31

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Introduction Prevalence of sling Multiple collisions KS vs. DNS

0.2 0.4 0.6 0.8 1.0 ωmax 4.1 8.2 12.3 16.4 20.5 ωrms 0.2 0.4 0.6 0.8 1.0 ωmax 1.2 2.4 3.5 4.7 5.9 ωrms

• M. Voßkuhle › Particle Collisions in Turbulent Flows

31

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Acknowledgments ∴ Conclusion References

• A. Pumir
• E. Lévêque
• J. Bec
• M. Lance
• B. Mehlig
• M. Reeks
• M. Wilkinson

Laboratoire de Physique

∞

• M. Voßkuhle › Particle Collisions in Turbulent Flows

32

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Acknowledgments ∴ Conclusion References

Conclusions and perspectives

› St > 0: Sling/caustics/RUM effect dominates

collision rates in turbulent flows

0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 Γ(a0)/Γ(2a0) St

› inertial particles may stay close for long times

10−6 10−4 10−2 100 102 0 1 2 3 4 5 6 7 8 9 10 ∆t1/TL 0.05 0.1 0.15 0.2 1 2 3 4 5 Γm/Γ1 St 20 40 60 80 100 ΓτK/(2a)3

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 2 4 6 8 10 P(Nc∣Nc ≥ 1) Nc St = 0.5 1.0 2.0 4.0

» leads to multiple collisions » overestimation of the collision kernel

M Voßkuhle et al. (2013a). “Multiple collisions in turbulent flows.” In: Phys. Rev. E 88, p. 063008 M Voßkuhle et al. (2013b). “Prevalence of the sling effect for enhancing collision rates in turbulent suspensions.” In: ArXiv e-prints (July 2013). arXiv: 1307.6853 [physics.flu- dyn] M Voßkuhle et al. (2011). “Estimating the Collision Rate of Inertial Particles in a Turbulent Flow: Limitations of the ‘Ghost Collision’ Approximation.” In: J. Phys.: Conf. Ser. 318, p. 052024

• M. Voßkuhle › Particle Collisions in Turbulent Flows

33

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Acknowledgments ∴ Conclusion References

Abrahamson J (1975). “Collision rates of small particles in a vigorously turbulent fluid.” In: Chem. Eng. Sci. 30, pp. 1371–1379. Andersson B et al. (2007). “Advective collisions.” In: EPL 80, p. 69001. Brunk BK, Koch DL, Lion LW (1998). “Turbulent coagulation of colloidal particles.” In: J. Fluid Mech. 364, pp. 81–113. eprint: http://journals.cambridge.org/article_S0022112098001037. Falkovich G, Fouxon A, Stepanov MG (2002). “Acceleration of rain initiation by cloud turbulence.” In: Nature 419, pp. 151–154. Fung JCH et al. (1992). “Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes.” In: J. Fluid Mech. 236, pp. 281–318. Gatignol R (1983). “Faxen Formulae for a Rigid Particle in an Unsteady Non-uniform Stokes Flow.” In: J. Méc. Théor. Appl. 1, pp. 143–160. Gustavsson K, Mehlig B, Wilkinson M (2008). “Collisions of particles advected in random flows.” In: New J. Phys. 10, p. 075014. Jullien M-C, Paret J, Tabeling P (1999). “Richardson Pair Dispersion in Two-Dimensional Turbulence.” In: Phys. Rev. Lett. 82, pp. 2872–2875. Lamb D (2001). “Rain Production in Convective Storms.” In: Meteorol. Monogr. 28,

• pp. 299–322.

Maxey MR, Riley JJ (1983). “Equation of motion for a small rigid sphere in a nonuniform flow.” In: Phys. Fluids 26, pp. 883–889.

• M. Voßkuhle › Particle Collisions in Turbulent Flows

34

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Acknowledgments ∴ Conclusion References

Monchaux R, Bourgoin M, Cartellier A (2010). “Preferential concentration of heavy particles: A Voronoï analysis.” In: Phys. Fluids 22.10. Rast MP, Pinton J-F (2011). “Pair Dispersion in Turbulence: The Subdominant Role of Scaling.” In: Phys. Rev. Lett. 107, p. 214501. Reeks MW, Fabbro L, Soldati A (2006). “In search of random uncorrelated particle motion (RUM) in a simple random flow field.” In: Proc. 2006 ASME Joint US European Fluids Engineering Summer Meeting. (July 17–20, 2006). Miami, FL, pp. 1755–1762. Saffman PG, Turner JS (1956). “On the collision of drops in turbulent clouds.” In: J. Fluid

• Mech. 1, pp. 16–30.

Scatamacchia R, Biferale L, Toschi F (2012). “Extreme Events in the Dispersions of Two Neighboring Particles Under the Influence of Fluid Turbulence.” In: Phys. Rev. Lett. 109, p. 144501. Shaw RA (2003). “Particle-Turbulence Interactions in Atmospheric Clouds.” In: Annu. Rev. Fluid Mech. 35, pp. 183–227. Simonin O et al. (2006). “Connection between two statistical approaches for the modelling of particle velocity and concentration distributions in turbulent flow: The mesoscopic Eulerian formalism and the two-point probability density function method.” In: Phys. Fluids 18, p. 125107. Sundaram S, Collins LR (1997). “Collision statistics in an isotropic particle-laden turbulent

• suspension. Part 1. Direct numerical simulations.” In: J. Fluid Mech. 335, pp. 75–109.
• M. Voßkuhle › Particle Collisions in Turbulent Flows

35

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Acknowledgments ∴ Conclusion References

Voßkuhle M, Pumir A, Lévêque E (2011). “Estimating the Collision Rate of Inertial Particles in a Turbulent Flow: Limitations of the ‘Ghost Collision’ Approximation.” In: J. Phys.:

• Conf. Ser. 318, p. 052024.

Voßkuhle M et al. (2013a). “Multiple collisions in turbulent flows.” In: Phys. Rev. E 88,

• p. 063008.

Voßkuhle M et al. (2013b). “Prevalence of the sling effect for enhancing collision rates in turbulent suspensions.” In: ArXiv e-prints (July 2013). arXiv: 1307.6853 [physics.flu-dyn]. Wang L-P, Wexler AS, Zhou Y (1998). “On the collision rate of small particles in isotropic

• turbulence. I. Zero-inertia case.” In: Phys. Fluids 10, pp. 266–276.

Wilkinson M, Mehlig B, Bezuglyy V (2006). “Caustic Activation of Rain Showers.” In: Phys.

• Rev. Lett. 97, p. 048501.
• M. Voßkuhle › Particle Collisions in Turbulent Flows

36

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Supplementary information Credits

Kinematic Simulations

u(x, t) =

Nk

∑

n=1

An cos(kn ⋅ x + ωnt) + Bn sin(kn ⋅ x + ωnt) An ⋅ kn = Bn ⋅ kn = 0 A2

n = B2 n = E(kn)∆kn,

E(kn) ∼ k−5/3

n

k1 = 2π L , kNk = 2π η , kn = k1 (L η)

(n−1)/(Nk−1)

ωn = λ √ k3

nE(kn),

λ ∶ “persistence parameter”

› efficient › highly “turbulent”

flows

› widely used

Fung et al. JFM (1992)

› Go back...

• M. Voßkuhle › Particle Collisions in Turbulent Flows

37

ÉCOLE NORMALE SUPÉRIEURE DE LYON

Supplementary information Credits

Bibliothèque municipale de Lyon. Photo: Vélo’Vs in snow. http://www.flickr.com/photos/ansobol, others from the corresponding websites. Photos and logos on the acknowledgements page. Lukaschuk S. Photo: Clustering particles. Dept. Engineering, Univ. Hull. http://commons.wikimedia.org. Some of the pictures are in the public domain and were taken from this website. “Rain”, “Clover” designed by Pavel Nikandrov, “Computer”, “Newspaper” designed by Yorlmar Campos, “Handshake” designed by Sam Garner,

• thers in the public domain. Icons from The Noun Project.
• M. Voßkuhle › Particle Collisions in Turbulent Flows

38