Aggregation and fragmentation dynamics for
a mixture of two types of inertial particles in
a turbulent flow
Ksenia Guseva
Ulrike Feudel
ICBM, University of Oldenburg
May 2012
M ARINE AGGREGATES
I
M.W Silver
Microaggregates: 100 − 500µm
I
Macroaggregates: > 500µm
I
Composed by: Algae,
Cyanobacteria, Bacteria,
inorganic and organic particles
R ICH COMPOSITION OF MARINE AGGREGATES
Dynamics of these aggregates is influenced by physical,
chemical and biological processes
inorganic material
(e.g. sand)
&
organic material
(e.g. algea)
O BJECTIVES
I
How the aggregation-fragmentation dynamics is
influenced by the densities of the particles involved.
I
I
Aggregates of a single type
aggregates of different types
T HE EQUATIONS OF MOTION
M AXEY-R ILEY EQUATIONS – I NERTIAL PARTICLES
dXt = V
dVt =
1
St (u − V) +
βDt u
X(t) — particle position
I
V(t) — particle velocity
I
Dt u = ∂ t u + u · ∇ u
I
u - synthetic turbulent flow
I
Dissipative range –
Kraichnan energy
2 2
spectrum: E(k) ∼ k3 e−λ k
0.20
30
0.15
25
0.10
20
0.05
15
0.00
0.05
10
0.10
5
0
I
0.15
0
5
10
15
20
25
30
0.20
I
I
λ – correlation length
τ – correlation time
PARAMETERS
M AXEY-R ILEY EQUATIONS
dVt =
dXt = V
1
St (u − V) +
3ρf
β=
.
ρf + 2ρp
βDt u
St =
r2
.
3βντ
I
ρp — particle density
I
r — particle radius
I
ρf — fluid density
I
I
β < 1 — heavy particles
ν — kinetic viscosity of the
flow
I
β > 1 — light particles
I
τ — time scale of the flow
I NTERACTING PARTICLES
A GGREGATION AND F RAGMENTATION
All aggregates are constituted of unbreakable unit particles.
The number of unit particles in an aggregate is α.
α i + α j → α i+j
α i+j → α i + α j
F RAGMENTATION
C RITICAL SHEAR
Particles break when the shear forces of the flow exceed the
binding forces.
∂uj
1 ∂ui
Scrit = γα−1/3
+
Sij =
2 ∂Xj
∂Xi
If Sflow (X) > Scrit → break.
S = (2Sij Sji )1/2
0.8
L
102
0.7
0.5
10−2 10−1 100 101
0.4
10−1
0.3
0.2
10−2
0.1
10−3 −5
10
10−4
10−3
10−2
S/ hSi
10−1
100
101
0
0
x
L
S
100
0.6
y
P (S)S
101
F RAGMENTATION
F RAGMENTS SIZE
Large-scale fragmentation
— two similar fragments
P(αnew)
P(αnew)
Erosion — shear forces act
closer to the edges
α
I
I
Results in binary fragmentation
Cascade of erosion processes
α
← δf rag →
C OLLISION RATES
S EGREGATION → COLLISION RATES
Heavy particles β < 1.
Light particles β > 1.
I
Light particles segregate to the area with high vorticity.
I
Heavy particles are expelled from these areas.
C OLLISION RATES Q( β 1 , β 2 )
M IXTURE OF UNIT PARTICLES WITH DENSITIES RATIOS β 1
Heavy particles β 1 and β 2 < 1.
0.6
β1 = 0.5
Q(β1 , β2 )
Q(β1 , β2 )
0.7
β1 = 0.3
0.025
0.020
0.015
0.5
β1 = 1.7
β1 = 2
β1 = 2.2
β1 = 2.4
0.4
0.3
0.010
0.2
0.005
0.1
0.000
β1
Light particles β 1 and β 2 > 1.
β1 = 0.1
0.030
AND
0.0
0.0
0.2
0.4
0.6
β2
0.8
1.0
1.6
1.8
2.0
β2
2.2
2.4
I
Collision rates are much higher for light particles than for
heavy particles.
I
The collision rates have a maximum for the equal particles.
A GGREGATION AND F RAGMENTATION
L IGHT PARTICLES
γ = 1.
30
25
α
20
15
10
5
0
10−1
3τ
4τ
5τ
time
6τ
7τ
8τ
γ = 2.
10−3
300
10−4
250
200
10−6
10−7
100
α
10−5
γ=1
γ=2
γ=5
101
150
100
50
0
α
102
3τ
103
α
Nα /N
10−2
4τ
time
5τ
900
800
700
600
500
400
300
200
100
2τ
3τ
4τ
6τ
7τ
time
8τ
9τ 10τ 11τ
6τ
A GGREGATION AND F RAGMENTATION
H EAVY PARTICLES
10−1
L
10−3
y
Nα /N
10−2
10−4
γ=8
γ=9
γ = 10
10−5
0
50
0
100
α
150
200
0
x
L
A GGREGATION AND F RAGMENTATION
H EAVY PARTICLES
100
β = 0.1
β = 0.5
1.95
10
1.90
−2
10−3
Nα /N
1.85
df
β = 0.1
β = 0.5
β = 0.8
10−1
1.80
10−4
10−5
1.75
10−6
1.70
10−7
10−8
1.65
0.0
0.5
1.0
St
1.5
2.0
0
20
40
α
St = 0.5
60
80
100
A GGREGATION AND F RAGMENTATION – M IXTURE
M ODEL
Fragmentation
I
I
The unit particles of both
types have the radius r
I
Consider homogeneous
aggregate
αnew1 = round
αp1 αnew δ
(αp1 + αp2 )
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αnew2 = αnew − αnew1
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δ is a random number
from 0 to 1
A GGREGATION AND F RAGMENTATION
H EAVY PARTICLES MIXTURE
β = 0.1, St = 0.5
β = 0.9, St = 4.5
β = 0.9, β2 = 0.1
10−1
10−2
Nα /N
10−3
10−4
10−5
10−6
10−7
0
20
40
α
60
80
100
C ONCLUSIONS AND FUTURE WORK
I
The fluid particle density ratio β influences the collision
rates between monomers.
I
The segregation of particles with distinct composition
decreases with the growth of aggregate size.
I
Influence of the fluid properties on
aggregation-fragmentation process