Turbulence Dispersion Force
– Physics, Model Derivation and Evaluation–
J.-M. Shi1, T. Frank2, A. Burns3
1 Institute of Safety Research, FZ Rossendorf
[email protected]
2 ANSYS CFX Germany
3 ANSYS CFX
FZR–ANSYS CFX Workshop on Multiphase Flow
29-30.06.2004, Dresden
FORSCHUNGSZENTRUM ROSSENDORF
Institut für Sicherheitsforschung
Contents
Physics
Modeling approaches
• Eulerian approach
• Lagrangian approach
A new model derivations
Evaluation
1
Turbulent dispersion (TD)
A result of the (dispersed phase) particle–eddy (continuous phase) interaction
Important for dispersed phase with small St
′
′
Turbulent mass diffusion (αk uk) and interphase momentum transfer
Turbulent mass diffusion is usually modeled by a mass diffusion term
The interphase momentum transfer due to turbulent dispersion is often
separated as an “interfacial force”, to be calculated from the temperal
correlation of the interfacial force fluctuations. Usually, only the drag
contribution is essential.
2
Eulerian model – mass diffusion
• Reynolds averaging
∂
(ρk αk ) + ∇ · (ρk αk Uk) = 0
∂t
′
applying φ = φ + φ
∂
′ ′
(ρk αk ) + ∇ · (ρk αk Uk) = −∇ · (ρk αk uk)
∂t
νk,t
′
′
(1)
⇒
with αk uk ∼ −
∇αk (2)
σk
• Adopting Favré-averaged velocity
′
′
α k uk
αk Uk
Ũk =
= Uk +
αk
αk
′′
′
Uk = Ũk + uk + uk,
′
′′
uk = −
(3)
′
α k uk
αk
6= 0
(4)
∂
(ρk αk ) + ∇ · (ρk αk Ũk) = 0
∂t
(5)
• Using Favré-averaged velocity simplifies the equation system
3
Eulerian model – turbulent dispersion force
The Favré-Averaged Drag Force (FAD) model,
Drag:
6 αf
1
FD = Cf p(Up − Uf ) = CD ρf |Up − Uf |
(Up − Uf )
|8
{z
} dp
| {z }
Af p
Df p
Reynolds averaging:

′
′
 α f uf
′
(6)
´
³
FD = Cf p Ũp − Ũf + FTD
′

αpuf  EDH
νf,t
FTD = Cf p 
−
 −→ Cf p
αf
αp
σf
Ã
∇αf
∇αp
−
αp
αf
(7)
!
(8)
For details refer to
A. Burns, T. Frank, I. Hamill and J.-M. Shi, ICMF2004, Paper No. 392
4
Lagrangian method
tracking trajectories of each particle, parcel
n
= xn
xn+1
p + Vp δt
p
Ã
!n
n+1
n
− Vp
Vp
3 ρf C D
1
=
|Vp − Vf | (Vp − Vf ) +
Fother
δt
4 ρp dp
ρpδV
(9)
(10)
construct fluid fluctuating velocity for particle-eddy interaction
n+1
,
t)
+
v
Vfn+1 = Uf (xn+1
p
f
Ã
!
δt
a = exp − ∗ ,
TL
b=
Ã
,
2kf
3
!1 q
2
1 − a2 ,
|e|n ∈ N (0, 1)
(11)
(12)
Lagrangian time step
P n+1
Fpn+1
F n+1
Pn = Fn
vfn+1 = avfn + ben , where
Eulerian time step
5
A new TD force model derivation (1)
Ensemble average for a cell △V at time tj
n(t )
n(t )
i
1 Xj i
1 Xj 3 CD
i − Vi |(Vi − Vi )δV i (13)
FD(tj ) =
ρf i |Vp
fD(tj ) =
p
f
f
△V i=1
△V i=1 4 dp
Time averaged drag
)
N n(t
Xj 3 C i
1 1 X
i − Vi |(Vi − Vi )δV iδt
FD =
ρf D
|
V
p
p
f
f
△V TE j=−N i=1 4 dip
(14)
TE (macro time scale) ≫ τe = C kǫ (eddy life time); δt ≪ τe
n(t−N )
n(t−j )
n(t0 )
n(tj )
n(tN )
△V
t−N = t − TE /2
t−j = t − jδt
t0 = t
tj = t + jδt
tN = t + TE /2
6
Model derivation (2)
Converting into the Eulerian frame
• Eulerian variables
n(t )
1 Xj
αp(tj ) =
δV i
△V i=1
(15)
Pn(tj ) i
Pn(tj ) i
i
Φ
δV
Φ δV i
1
i=1
i=1
Φ(tj ) = P
=
n(tj )
△V
αp(tj )
i
i=1 δV
(16)
• Reynolds averaging operator
N
1 t+TE /2
1 X
Φ(tj )δt
Φ(t) =
Φ(θ)dθ =
TE t−TE /2
TE j=−N
Z
(17)
• Averaging the drag force
FD = αpfD =
αpfD
| {z }
mean drag
+
′
α′pfD
| {z }
turbulent dispersion force
(18)
7
Derivation details (1)
¢
¡
3 CD
|Up − Uf | Up − Uf αp
ρf
4 dp
¢
¢
¡
3 CD ¡
| Up − Uf | αp Up − Uf
ρf
4 dp
FD =
≈
|
=
=
=
=
{z
D
}
D|Up − Uf | αpUp − αpUf
D|Up − Uf |

³
µ
′
′
αpŨp − αp Uf − αpuf
´
¶
′

αp ′ ′
′ ′
Dαp|Up − Uf | Ũp − Ũf + D|Up − Uf |
α f uf − α p uf
αf
!

D|Up − Uf | αp
³
´
³
|
′
´
Ũp − Ũf + αp
Ã
′
α f uf
αf
′

− α p uf 
(19)
{z
}
turbulent dispersion force FTD
8
Derivation details (2)
Eddy viscosity hypothesis (EVH)
νf,t
′ ′
α uf = −
∇α
σf
′′
uf =
⇒
νf,t
σf α
∇α
(20)
Model expression (for the continuous phase)
Ã
αp
3 C νf,t
FTD ≈ ρf D
|Up − Uf | ∇αp −
∇αf
4 dp σf
αf
|
{z
CT D
!
(21)
}
Results for two-fluid model, αp + αf = 1
∇αp
3 CD νf,t
|Up − Uf |
FTD ≈ ρf
4 dp σf
αf
(22)
9
Remarks
The present derivation illustrates the physics of the turbulent dispersion
The present derivation explains why double average makes sense
Lagrangian evaluation of turbulent dispersion is very expensive. This derivation might provide a theoretical foundation for a deterministic TD
force model for the Lagrangian solver
10
Evaluation
11
FZR MTLoop test facility
Air-water system, isothermal
Inner pipe diameter D = 51.2 mm
Wire mesh sensor measurements
Test section from gas injection:
L = 0.03 to 3.03 m
Injection nozzle arrangement:
12
Test case definition
13
Two-fluid model Evaluation
1.8
2.5
FZR-038 experiment
FAD-SST
RPI-SST, CTD = 0.35
1.6
1.4
FZR-039 experiment
FAD-SST
RPI-SST, CTD = 0.35
2
1.5
1
αg∗
αg∗
1.2
0.8
1
0.6
0.4
0.5
0.2
0
0
0.2
0.4
0.6
r∗
0.8
1
0
0
0.2
0.4
0.6
0.8
1
r∗
For details refer to
J.-M. Shi, T. Frank, E. Krepper, D. Lucas, U. Rohde, H.-M. Prasser,
ICMF2004, Paper No.400
Th. Frank, J.-M. Shi, and A. Burns, 3rd International Symposium on
Two-Phase Flow Modeling and Experimentation, Pisa, Italy, 22-24 September, 2004
14
Poly-dispersed model Evaluation
• Stationary, axisym., bubbly flow at the upper test section (L/D = 59.2)
• Data from measurements: superficial velocities, mean bubble diameter,
local gas volume fraction
Ul
Ug
Air
Air 1
Air 2
Air 3
[m/s] [m/s] VF[%] dp VF[%] dp VF[%] dp VF[%]
070 0.161 0.0368 22.86 4.8 12.20 7.0 10.66
071 0.255 0.0368 14.43 4.8 11.27 6.6
3.16
072 0.405 0.0368
9.09
4.6
8.18
6.4
0.91
072 0.405 0.0368
9.09
3.8
2.48
5.0
5.70
6.4
0.91
083 0.405 0.0574 12.76 4.8
9.86
6.7
2.90
083 0.405 0.0574 12.76 3.7
1.00
5.0
8.86
6.7
2.90
084 0.641 0.0574
8.95
4.6
8.24
6.4
0.71
107 1.017 0.140
13.77 5.1
9.47
6.8
4.30
108 1.611 0.140
8.69
4.7
8.10
6.3
0.59
110 4.047 0.140
3.46
2.7
3.46
110 4.047 0.140
3.46
2.4
1.71
3.4
1.75
Ul, Ug–superficial velocity, dp–diameter [mm], VF–gas volume fraction.
Index
15
Results from poly-dispersed model
2
1
0.5
0
0
2
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
∗
r
2
FZR-083 exp.
sum
d= 3.7mm
d= 5.0mm
d= 6.7mm
1
0.5
0.6
0.8
1
0.6
0.8
1
∗
FZR-084 exp.
sum
d= 4.6mm
d= 6.4mm
1.5
αg∗
1.5
αg∗
1
0.5
r
0
FZR-071 exp.
sum
d= 4.6mm
d= 6.4mm
1.5
αg∗
1.5
αg∗
2
FZR-070 exp.
sum
d= 4.8mm
d= 7.0mm
1
0.5
0
0.2
0.4
0.6
r∗
0.8
1
0
0
0.2
0.4
r∗
16
TD coefficient, FZR070, CT D =
14
10
8
6
4
0
0.1
0.2
0.3
0.4
0.5
r
2.5
d= 4.8mm,
d= 7.0mm,
d= 4.8mm,
d= 7.0mm,
2
CD, Slip Velocity
10
8
6
4
0.6
0.7
0.8
0
0.1
0.2
0.3
∗
0.4
0.5
r
2.5
CD, FZR070Grace
CD
Slip V
Slip V
d= 4.8mm,
d= 7.0mm,
d= 4.8mm,
d= 7.0mm,
2
1.5
1
0.5
0
2
0.9
CD, Slip Velocity
2
− Uf |
d= 4.8mm, CTD, FZR070Ishii&Zuber
d= 6.6mm, CTD
12
Turb. Disp. Coeff
12
Turb. Disp. Coeff
14
d= 4.8mm, CTD, FZR070, Grace
d= 6.6mm, CTD
3 ρ CD νf,t |U
p
4 f dp σf
0.6
0.7
0.8
0.9
0.8
0.9
∗
CD, FZR070Ishii&Zuber
CD
Slip V
Slip V
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
r
∗
0.6
0.7
0.8
0.9
0
0
0.1
0.2
0.3
0.4
0.5
r
0.6
0.7
∗
17
TD coefficient, FZR110, CT D =
35
35
d= 2.4mm, CTD, FZR110, Grace
d= 3.4mm, CTD
25
20
15
0
0.1
0.2
0.3
0.4
0.5
r
1
d= 2.4mm,
d= 3.4mm,
d= 2.4mm,
d= 3.4mm,
0.8
CD, Slip Velocity
d= 2.4mm, CTD, FZR110Ishii&Zuber
d= 3.4mm, CTD
25
20
15
0.6
0.7
0.8
0
0.1
0.2
0.3
∗
0.4
0.5
r
1
CD, FZR110, Grace
CD
Slip V
Slip V
d= 2.4mm,
d= 3.4mm,
d= 2.4mm,
d= 3.4mm,
0.8
0.6
0.4
0.2
0
10
0.9
CD, Slip Velocity
10
− Uf |
30
Turb. Disp. Coeff
Turb. Disp. Coeff
30
3 ρ CD νf,t |U
p
4 f dp σf
0.6
0.7
0.8
0.9
0.8
0.9
∗
CD, FZR110Ishii&Zuber
CD
Slip V
Slip V
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
r
∗
0.6
0.7
0.8
0.9
0
0
0.1
0.2
0.3
0.4
0.5
r
0.6
0.7
∗
18