Post on 09-Jul-2018
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
A Coupled VOF-Eulerian Multiphase CFD Model To SimulateBreaking Wave Impacts On Offshore Structures
Pietro Danilo Tomaselli
Ph.d. student
Section for Fluid Mechanics, Coastal and Maritime Engineering
Department of Mechanical Engineering
Technical University of Denmark
24th February 2016
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Outline
Introduction
Model set-up
Description of the model
Validation of the model
Application of the model
Conclusions
References
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Introduction
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Offshore wind farms in the future
Multi-use platform (wind farms, aquaculture and exploitation of wave energy)
Massive development in the intermediate depth region (20 - 60 m)
EU Project ’MERMAID - Innovative Multi-purpose offshore platforms: planning, design and operation’.
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Spilling breaking waves impact on secondary structures
Waves often break as spilling breakers in the intermediate depth area under stormconditions
Spilling waves are characterized by a mixture of dispersed air bubbles and watertraveling with the wave front
Impact on secondary structures (external access platforms, boat-landings, railings..)can cause severe damages
Spilling and plunging waves Breaking waves impact at a Horns Rev wind turbine
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
The study
Which is the scope?Development of a CFD solver that can simulate the air entrainment inbreaking waves
How is it realized?The solver is built by the means of the open source package OpenFOAM
What is it applicable for?
- The solver is optimized for spilling breaking waves
- The compressibility of the air is neglected
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Model set-up
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Which is the main problem from the numerical point of view?
Wave breaking is an unsteady multiphase flow where a wide range of interfaciallength scales is involved
air
water
air bubbles
Wave propagation → O(m)! → scales larger thanthe grid size → Volume Of Fluid
Breaking event: entrainment of air bubbles →O(10−4 m)! → scales smaller than the grid size→ method for dispersed flow
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Which method to handle the motion of bubbles?
Balachandar,2009:
St =τpτξ
=
(
2ρ+ 1
36
)(
1
1 + 0.15Re0.687p
)(
d
ξ
)2(ξ
η
)4/3
Assuming: ρ = 0.001, ξ = 0.01 m, η ≈ 1 · 10−5 m, d = [0.0006 0.0008 0.0010 0.00120.0016 0.0020 0.0025 0.0031 0.0039 0.0049 0.0062 0.0078 0.0099] m
100
101
102
103
0.001
0.2
1
d/η [-]
St[-]
Dusty gas
Equilibrium Eulerian
Eulerian
Lagrangian
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
A quick look at the multiphase Eulerian methodology
N-S equations discretized in a control volume with n phases inside → n equations forthe bulk of the volume + n − 1 jump conditions across interfaces among phases.
Equations are spatially averaged:
- phase fraction α stems
- the momentum transfer at the interface is not resolved and then needs to be modeledby a sub-grid term
∂(αiρi)
∂t+∇ · (αiρiui) = Si
∂(αiρiui)
∂t+∇ · (αiρiui ⊗ ui) = −αi∇p +∇ · (αiρiTi) + αiρig +Mi
Set of equations solved per each phase → 2n equations
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
How can the coupling be realized?
Coupling means that the VOF-like solver is combined with the multiphase Eulerianmethodology.
Which phases are involved?water + air above the free surface + n bubble classes = n+2!
Different couplings exist, two have been tried:
air bubbles
air
water
Eulerian framework
for all phases
Numerical sharpening and Mi
air bubbles
mixture
Eulerian for bubbles in the mixture
VOF for the mixture
Numerical sharpening
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
VOF with momentum transfer modeling at the free surface
”Glass of water” Propagation of a solitary wave
−4 −2 0 2 4 6 8
0
0.03
0.06
0.09
0.12
0.15
Surface
elevationfrom
SW
L[m
]x [m]
Instabilities at the free surface in both cases!
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
VOF without momentum transfer modeling at the free surface
water (w)continuous air (air)n classes
ρmixt =αwρw+αairρair
αw+αair
νmixt =αwνw+αairνair
αw+αair
n classesmixture (mixt)
Calculation of pressureand velocities (PISO algorithm)
Calculation of void fractions(MULES algorithm)
”Glass of water”
Main achievement: modeling of the momentum transfer between water andcontinuous air not needed anymore!
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Description of the model
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Governing equations
Averaged mass and momentum conservation equations:
∂(αiρi)
∂t+∇ · (αiρiui) = Si
∂(αiρiui)
∂t+∇ · (αiρiui ⊗ ui) = −αi∇p +∇ · (αiρiT
effi ) + αiρig +Mi
The n classes have the same density and viscosity but they have different diameter
Si and Mi represent mass transfer and interfacial forces among phases respectively
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Effective stress tensor Teffi
By employing the eddy-viscosity approximation, the effective stress tensor is:
Teff = 2(νeffi ])
[
1
2(∇ui +∇uT
i )−1
3(∇ · ui)I
]
−2
3kiI
The effective viscosity of the mixture phase νeff is (Deen et al., 2001):
νeff = ν + νt + νBI
ν = νmixt and νt is the turbulent viscosity calculated by a dynamic Smagorinsky model
νBI is an extra-term accounting for the bubble-induced turbulence (Sato et al., 1975):
νBI = 0.6
n∑
i=1
αidi |ui − umixt |
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Mass exchange among phases Si
Si = B+i + B−
i + C+i + C−
i + Ei + Di
Mass transfer among the n classes:- breakage → B+
i + B−i (Prinche et Blanch, 1990)
- coalescence → C+i + C−
i (Martinez-Bazan et al., 1999)
dj < di < dk
breakage
coalescence
Mass transfer between the dispersed bubbles and air-from air into the n classes → air entrainment → Ei
-from the n classes into air → degassing → Di
Degassing modeled as (Hansch et al., 2012):
Di = ϕairρiαi/ (at∆t)
where ∆t is the time step, at = 20 a constant and ϕair = 0.5tanh[100(αair − 0.5)] + 0.5
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Air entrainment modeling
The air entrainment is reproduced by a sub-grid scale model (Derakhti and Kirby,2014):
Ei =cen
4π
ρmixt
σαmixt
(
f∆i∑n
i (di2 )
2f∆i
)
ǫsgs
- cen is a parameter which regulates the amount of entrained bubbles- fi is the size spectrum of the entrained bubbles (Deane and Stokes, 2002):
fi =
{
(di2)−
103 if (di
2) > 1 mm.
(di2 )− 3
2 if (di2 ) ≤ 1 mm.
- ∆i is the width of each bubble class
Bubbles are entrained at the free surface cells when ǫsgs is larger than a fixedthreshold!
ǫsgs is the rate of transfer of energy from the resolved to the sub-grid scales modeled byLES turbulence model:
ǫsgs = 2νt‖0.5(∇umixt +∇uTmixt)‖
2
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Momentum transfer among phases Mi
Between every class and mixture phase:
Mi = MD,i +ML,i +MVM ,i +MTD,i
- Drag force MD,i =34ρmixtαmixtαiCD
‖ui−umixt‖(ui−umixt)di
- Lift force ML,i = −ρmixtαmixtαiCL(ui − umixt)× (∇× umixt)
- Virtual mass force MVM ,i = ρmixtαmixtαiCVM
(
Dumixt
Dt− Dui
Dt
)
- Turbulent dispersion force MTD,i = −34CD
ρmixt
di
(νt+νBI )Sb
‖ui − umixt‖∇αi
Between water and continuous air, i.e. at the free surface, the tension force isaccounted as (Brackbill et al., 1992):
Msurf ,i = σκ∇α (1)
σ = 0.0728 N/m and κ is the free surface curvature.
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Free surface sharpening method
An additional term is added to the interface transport equations of water andcontinuous air:
∂(αiρi)
∂t+∇ · (αiρiui) +∇ · [ucαi(1− αi)] = Si
The ”compression velocity”uc compresses the interface counteracting the numericaldiffusion:
uc = min(C‖ur‖,max(‖ur‖))∇α
‖∇α‖
C is a coefficient that the user can specify and was taken as 1
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Validation of the model
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Bubble column of (Deen et al., 2001): simulation set-up
z
x
y
0.15 m
0.15
m
0.4
5 m
1.00 m
air
measurement point
(0,0,-0.20m)
init
ial w
ate
r le
vel
Uniform 3D hexaedral mesh of size 0.01 m
LES dynamic Smagorinsky model employed
Flow simulated for 600 s
Turbulence quantities time-averaged after the first 30 s.
Two different number of classes n:- n = 1 → di = 4 mm- n = 11 → di = [1.0 1.2 1.6 2 2.5 3.1 4.0 5.0 6.3 8 10]mm
Why this case? It resembles the motion of the bubbleplume in waves. The difference is that the air entrainmentis imposed by boundary conditions.
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Bubble column of (Deen et al., 2001): results
−0.075−0.05−0.025 0 0.025 0.05 0.075−0.1
0
0.1
0.2
0.3
x [m]
uz,w
[ms−
1]
experimentn = 1n = 11
Mean water axial velocity
−0.075−0.05−0.025 0 0.025 0.05 0.075−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [m]
√
u′2x,y,z,w
[ms−
1]
experiment√
u′2x,w
√
u′2y,w
√
u′2z,w
Mean water turbulence fluctuations
−0.075−0.05−0.025 0 0.025 0.05 0.0750
0.1
0.2
0.3
0.4
0.5
x [m]uz,air[m
s−1]
experimentn = 1
Mean air axial velocity
−0.075−0.05−0.025 0 0.025 0.05 0.075−0.01
0
0.01
0.02
0.03
x [m]
kw[m
2s−
2]
experimentn = 1n = 11
Mean turbulent kinetic energy
−0.075−0.05−0.025 0 0.025 0.05 0.0750
0.01
0.02
0.03
0.04
0.05
x [m]
αair[-]
n = 1n = 11
Mean gas hold-up
10−2
100
102
104
10−15
10−10
10−5
100
105
Frequency [Hz]
Spe
ctra
l den
sity
[m2 s
]
−5/3
−3
u′
x,w
u′
y,w
u′
z,w
Spectrum of turbulence
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Bubble column of (Deen et al., 2001): the flow
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Application of the model
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated unsteady spilling wave
Case set-up:
- Flume is 25 m long by 0.6 wide by 0.6 m high. Still water level at 0.4 m from theconstant bottom
- Breaking event by a dispersive focusing method:80 linear components of a JONSWAPspectrum (Tp = 1.7 s, Hs = 0.084 m, γ = 3.3.)
- Linear superposition at focusing point x = 15.0 m
Simulation:
- Quasi-uniform 3D mesh of size 0.0125 m
- Flow simulated for 25 s.
- LES dynamic Smagorinsky employed
- waves2Foam (Jacobsen et al., 2012) coupled with the model and used forwave generation
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated unsteady spilling wave: surface elevation
−5 −4 −3 −2 −1 0 1 2 3−0.2
−0.1
0
0.1
0.2
η[m
]
t∗ [-]
−5 −4 −3 −2 −1 0 1 2 3−0.2
−0.1
0
0.1
0.2
η[m
]
t∗ [-]
linearCFD
linearCFD
Upper: x=8 m. Lower: x = xb = 15.0 m
Simulated breaking point xb ≈ 15.0 m
Simulated breaking time tb ≈ 16.85 s
Period of highest wave Tc ≈ 1.6 s
t∗ =t − tob
Tc
Comparison reveals second order effects
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated unsteady spilling wave: bubble entrainment
Parameters cen and ǫsgs of the air entrainment model need to be calibrated. How?(Lamarre and Melville, 1991) measured void fraction in similar waves → cen = 20 andǫsgs = 0.01 m2 s−3
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
1.2
Vb /V
0 [−]
t∗ [-]
0 0.5 1 1.50
10
20
30
Ab /V
0 [−]
t∗ [-]
experiment7phases14phases
experiment7phases14phases
- Two simulations: 7 phases and 14 phases
- Volume per unit length of crest:
V b =
∫
A
αbH(αb − αbthld)dA
- Cross-sectional area:
Ab =
∫
A
H(αb − αbthld)dA
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated unsteady spilling wave: the bubble plume
As in study of (Rojas and Loewen, 2010), the roller moved downstream with a speed ≈100% of the celerity of the highest wave (Cc = Lc/Tc ≈ 1.8 m s−1) and the thicknessof the roller was around 5 cm.
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated spilling wave: impact on a cylinder
In the same domain a (slender) cylinder with a diameter D = 0.05 was placed at x =15.7 m (why? to let the plume develop!)
Computation of computed of in-line force without and with bubbles (7 phases)
No-slip condition for water velocity at the cylinder surface
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated spilling wave: impact without bubbles
Comparison with the Morison force:
- the cylinder center x = 15.7 m used for analytical surface elevation, velocity andacceleration
- CD and CM evaluated according to Re and KC at different depths
- Based on LWT: 10 / Re / 3 · 104 and KC ≈ 20
−3 −2 −1 0 1 2 3−0.2
−0.1
0
0.1
0.2η[m
]
t∗ [-]
−3 −2 −1 0 1 2 3−6
−4
−2
0
2
4
6
In−
line
For
ce [N
]
t∗ [-]
linearCFD
MorisonCFD
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated spilling wave: impact with bubbles
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
An isolated spilling wave: impact without VS with bubbles
Differences are recognized at the interval 0.15 ≤ t∗ ≤ 0.4 when the bubble plumepassed over the cylinder
Largest differences are localized at around t = 0.25 t∗ when the volume of the bubbleplume was maximum
0 0.25 0.5 0.75 1 1.25 1.5−6
−4
−2
0
2
4
6In
−lin
e F
orce
[N]
t∗ [-]
0.1 0.15 0.2 0.25 0.3 0.35 0.4−6
−4
−2
0
2
4
6
In−
line
For
ce [N
]
t∗ [-]
CFD no bubblesCFD with bubbles
CFD no bubblesCFD with bubbles
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
The boundary layer around the cylinder
In the adopted model, the cell size must be larger the biggest bubble (≈ 1 cm usually)
For the highest Re - just near the roller! - separation and vortex shedding occur
The resolution of mesh around the cylinder may not be enough high to capture theseparation
- uniform current in the 3D flume
- Re = 2 · 104
- CD ≈ 1.3
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3
t [s]
CD[-]
dynamic Smagorinskydynamic Smagorinsky with WFconstant Smagorinsky with vDDES
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Conclusions
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Main conclusions and on-going work
A CFD simulation of the entire breaking wave process involves interfacial length scalesboth smaller and larger than the grid size.
An Eulerian model coupled with a VOF-type interface capturing algorithm wasdeveloped to handle such multi-scale problem.
A bubble column flow was analyzed to test the momentum transfer modeling and theimplemented mass transfer formulations.
An isolated breaking wave generated by a dispersive focusing method was simulated totest the air entrainment model
The impact of the spilling wave on a slender cylinder was reproduced to estimate theeffect of bubbles on the exerted in-line force.
Further investigations are needed, especially concerning the flow around the cylinder.
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
References
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
References
References I
Balachandar S. 2009. A scaling analysis for point-particle approaches to turbulentmultiphase flows. Int. J. Multiphase Flow 35:801-10
Sato, Y., and Sekoguchi, K., 1975. ”Liquid velocity distribution in two-phase bubbleflow”. International Journal of Multiphase Flow, 2(1), pp. 79-95.
Martinez-Bazan, C., Montanes, J., and Lasheras, J., 1999. ”On the breakup of an airbubble injected into a fully developed turbulent flow. Part 1. Breakup frequency”.Journal of Fluid Mechanics, 401, pp. 183-207.
Martinez-Bazan, C., Montanes, J. L., and Lasheras, J., 1999. ”On the breakup of an airbubble injected into a fully developed turbulent flow. Part 2. Size PDF of the resultingdaughter bubbles”. Journal of Fluid Mechanics, 401, pp. 183-207.
Prince, M. J., and Blanch, H. W., 1990. ”Bubble Coalescence and Break-Up inAir-Sparged Bubble Columns”. AIChE Journal, 36(10), pp. 1485-1499.
Hansch, S., Lucas, D., Krepper, E., and Hohne, T., 2012. ”A multi-field two-fluidconcept for transitions between different scales of interfacial structures”. InternationalJournal of Multiphase Flow, 47, Dec., pp. 171-182.
DTU 2016 Technical University of Denmark
Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
References
References II
Derakhti, M., and Kirby, J. T., 2014. ”Bubble entrainment and liquid-bubble interactionunder unsteady breaking waves”. Journal of Fluid Mechanics, 761, Nov., pp. 464-506.
Deane, G. B., and Stokes, M. D., 2002. ”Scale dependence of bubble creationmechanisms in breaking waves.”. Nature, 418(6900), Aug., pp. 839-44.
Brackbill, J., Kothe, D., and Zemach, C., 1992. ”A continuum method for modelingsurface tension”. Journal of Computational Physics, 100, pp. 335-354.
Deen, N. G., Solberg, T., and Hjertager, B. H., 2001. ”Large eddy simulation of theGas Liquid flow in a square cross-sectioned bubble column”. Chemical EngineeringScience, 56, pp. 6341-6349.
Rojas, G., and Loewen, M. R., 2010. ”Void fraction measurements beneath plungingand spilling breaking waves”. J. Geophys. Res., 115(C8), p. C08001.
DTU 2016 Technical University of Denmark