Reynolds number scaling of inertial particle statistics in
turbulent channel flows
Matteo Bernardini
Dipartimento di Ingegneria Meccanica e Aerospaziale
Università di Roma “La Sapienza”
Paolo Orlandi’s 70th birthday
VORTICAL STRUCTURES AND WALL TURBULENCE
September 19-20, 2014
Roma
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Motivations
Turbulent flows laden with solid particles common in nature and technology
Transport and sedimentation processes
Atmospheric dispersion of pollutants
Particles droplets in steam turbines
Transport of chemical aerosols
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Inertial particles in wall-bounded flows
Preferential concentration (particle clustering)
Inertial particles cannot follow the instantaneous fluid flow streamlines
Low particle concentration found in the cores of strong vortices
Turbophoretic drift
Strong segregation of particles in the near wall-region
Net mass transfer against a gradient in turbulence intensity
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Inertial particles in wall-bounded flows
Preferential concentration (particle clustering)
Inertial particles cannot follow the instantaneous fluid flow streamlines
Low particle concentration found in the cores of strong vortices
Turbophoretic drift
Strong segregation of particles in the near wall-region
Net mass transfer against a gradient in turbulence intensity
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Turbophoresis
Turbophoretic drift
Preferential concentration in the spanwise direction
Particle transfer driven by the action of strong ejections and sweeps (Marchioli
& Soldati, JFM 2002)
Only intermediate-size particles affected by turbophoresis
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Turbophoresis
Turbophoretic drift
Preferential concentration in the spanwise direction
Particle transfer driven by the action of strong ejections and sweeps (Marchioli
& Soldati, JFM 2002)
Only intermediate-size particles affected by turbophoresis
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
4 / 16
Background and objectives
Background
Experimental difficulties (high particle concentration in the viscous sublayer)
Most analysis made using DNS coupled with Lagrangian particle tracking
Previous studies limited to low Reynolds numbers (Reτ = 150 − 300)
Main goal
Understand the effect of the Reynolds number
Key open questions:
is the turbophoretic drift universal with the Reynolds number?
what’s the effect of the Reynolds number on the particle deposition rate?
DNS up to Reτ = 1000 !
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
5 / 16
Background and objectives
Background
Experimental difficulties (high particle concentration in the viscous sublayer)
Most analysis made using DNS coupled with Lagrangian particle tracking
Previous studies limited to low Reynolds numbers (Reτ = 150 − 300)
Main goal
Understand the effect of the Reynolds number
Key open questions:
is the turbophoretic drift universal with the Reynolds number?
what’s the effect of the Reynolds number on the particle deposition rate?
DNS up to Reτ = 1000 !
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
5 / 16
Background and objectives
Background
Experimental difficulties (high particle concentration in the viscous sublayer)
Most analysis made using DNS coupled with Lagrangian particle tracking
Previous studies limited to low Reynolds numbers (Reτ = 150 − 300)
Main goal
Understand the effect of the Reynolds number
Key open questions:
is the turbophoretic drift universal with the Reynolds number?
what’s the effect of the Reynolds number on the particle deposition rate?
DNS up to Reτ = 1000 !
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
5 / 16
Numerical methodology
Carrier phase
Solution of NS equations for a solenoidal velocity field
Second-order FD solver on staggered mesh (provided by Orlandi)
Discrete kinetic energy preservation
Simulations performed in convecting frame (Bernardini et al. 2013)
Doubly-Cartesian MPI splitting (x-z → best choice for load-balance)
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Numerical methodology
Particle phase
Particle equations
Diluted dispersion of pointwise, spherical particles → one-way coupling
Heavy particles, density ratio ρp /ρf ≈ 2700 (sand in air)
Motion of a small rigid sphere described by Maxey and Riley (PoF 83)
Forces neglected: added mass, fluid acceleration, Basset history force, gravity
dxp
= up
dt
dup
kp
=
(u − up )
dt
τp
(1)
Particle relaxation time τp = ρp dp2 /18µ (measure of inertia)
Stokes number St = τp /τf → St = τp u2τ /ν
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Physical and computational parameters
Flow case
P150
P300
P550
P1000
Reτ
145
299
545
995
Nx
384
768
1280
2560
Ny
128
192
256
512
Nz
192
384
640
1280
∆x+
7.1
7.3
8.0
7.3
∆z+
4.7
4.9
5.3
4.8
Tuτ /h
657
410
326
170
T+
95265
122590
177670
169150
Table : List of parameters for particle-laden turbulent channel flows.
St
1
10
25
100
500
1000
dp+
0.082
0.25
0.41
0.82
1.83
2.58
dp (µm)
1.0
3.2
5.0
10.0
22.4
31.6
τp (s)
9.55 · 10−6
9.55 · 10−5
2.39 · 10−4
9.55 · 10−4
4.78 · 10−3
9.55 · 10−3
ρp /ρf
2700
2700
2700
2700
2700
2700
Table : Relevant parameters for the inertial particles
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Analysis of particle dispersion
Shannon entropy
How quantify the wall accumulation process?
Shannon entropy parameter: single global indicator
Entropy parameter estimation
Slice the box in M slabs
Compute pj = Nj /N, j = 1, . . . , M
Entropy: S =
−
M
j=1
pj log pj
log M
S = 0 → particles into a single slab
S = 1 → uniform distribution
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
9 / 16
Analysis of particle dispersion
Shannon entropy
How quantify the wall accumulation process?
Shannon entropy parameter: single global indicator
Entropy parameter estimation
Slice the box in M slabs
Compute pj = Nj /N, j = 1, . . . , M
Entropy: S =
−
S=0
M
j=1
S=1
pj log pj
log M
S = 0 → particles into a single slab
S = 1 → uniform distribution
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
9 / 16
Particle dispersion (Shannon entropy evolution)
P150
P300
1.2
1.2
1
S
0.8
0.6
1
0.8
S
St = 1
St = 10
St = 25
St = 100
St = 500
St = 1000
0.6
0.4
0.4
0.2
0.2
0
0
50
100
0
150
0
50
t+ (×10−3 )
100
150
t+ (×10−3 )
P550
P1000
1.2
1
1
0.8
0.8
0.6
0.6
S
S
1.2
0.4
0.4
0.2
0.2
0
0
50
100
+
t (×10
−3
150
0
50
100
150
t+ (×10−3 )
)
Matteo Bernardini (Università di Roma La Sapienza)
0
Particle-laden channel flows
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Wall-normal concentration profiles
0
10
St = 1
-3
10
-4
10
-5
10
St = 10
10
0
10
1
10
2
10
3
10
-3
10
-4
10
-5
10
0
10
1
y
2
10
3
-3
10
-4
10
-5
10
10
St = 100
10
St = 500
c+
c+
10
-3
10
-3
10
10
-4
10
-4
10
-5
10
-5
10
-5
2
10
3
y
10
3
St = 1000
10-2
-4
10
2
10-1
-3
+
10
0
10
1
1
y
0
10-2
10
10
+
10-1
10-2
0
0
y
10-1
c+
10
10
+
0
10
St = 25
10-2
+
10
0
10-1
10-2
c+
c+
10-2
10
0
10-1
c+
10
10-1
10
0
10
1
10
2
10
3
+
y
10
0
10
1
10
2
10
3
y+
Lines: Reτ = 150; Reτ = 300; Reτ = 550; Reτ = 1000.
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Spatial organization
Instantaneous distribution of St = 25 particles at Reτ = 1000
Spanwise spacing of the particle streaks at (St = 25)
200
∆p z+
150
100
50
0
0
200
400
600
800
1000
1200
Reτ
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Deposition rate
Deposition coefficient
The rate at which particle deposit is fundamental for practical purposes
particle mass transfer rate at the wall J = dN/dt/Ad
bulk concentration C = N/φ
In the case of channel flow
kd = −J/C =
h dN
N dt
(2)
kd has dimension of velocity
kd+ =
Matteo Bernardini (Università di Roma La Sapienza)
1 dN h
N dt uτ
Particle-laden channel flows
(3)
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Deposition coefficient
10
1
Reτ
100
kd+
10-1
10
-2
10
-3
10
-4
10
0
10
1
10
2
10
3
St
Lines: Reτ = 150; Reτ = 300; Reτ = 550; Reτ = 1000.
kd+ =
1 dN h
N dt uτ
Matteo Bernardini (Università di Roma La Sapienza)
dt = dt+
δv
uτ
Particle-laden channel flows
→
kd+
1 dN
=
Reτ
N dt+
(4)
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Deposition coefficient
10
1
Reτ
100
kd+
10-1
10
-2
10
-3
10
-4
10
0
10
1
10
2
10
3
St
Lines: Reτ = 150; Reτ = 300; Reτ = 550; Reτ = 1000.
kd+ =
1 dN h
N dt uτ
Matteo Bernardini (Università di Roma La Sapienza)
dt = dt+
δv
uτ
Particle-laden channel flows
→
kd+
1 dN
=
Reτ
N dt+
(4)
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Deposition coefficient
Inner-scaling
replacements
Deposition rate of segregated particles collapse in inner units
10
1
kd+ /Reτ
100
10-1
10
-2
10
-3
10
-4
10
0
10
1
10
2
10
3
St
Lines: Reτ = 150; Reτ = 300; Reτ = 550; Reτ = 1000.
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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Conclusions
DNS of particle-laden turbulent channel flows up to Reτ = 1000
Particle St numbers in the range St = 1 − 1000
Universality of the turbophoretic drift
Particle concentration curves collapse in inner-scaling
Spanwise spacing of particle streaks constant with Reτ
Inner-scaling to collapse the deposition rate of segregated particles
Reference
M. Bernardini, Reynolds number scaling of inertial particle statistics in turbulent
channel flows, JFM-RP 2014, in press
Acknowledgments
Thanks to P. Orlandi, S. Pirozzoli and F. Picano
Matteo Bernardini (Università di Roma La Sapienza)
Particle-laden channel flows
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