A Coupled VOF-Eulerian Multiphase CFD Model To Simulate Breaking Wave Impacts On Offshore Structures Pietro Danilo Tomaselli Ph.d. student Section for Fluid Mechanics, Coastal and Maritime Engineering Department of Mechanical Engineering 24th February 2016
Outline Introduction Model set-up Description of the model Validation of the model Application of the model Conclusions References
Introduction
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.
Spilling breaking waves impact on secondary structures Waves often break as spilling breakers in the intermediate depth area under storm conditions Spilling waves are characterized by a mixture of dispersed air bubbles and water traveling 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
The study Which is the scope? Development of a CFD solver that can simulate the air entrainment in breaking 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
Model set-up
Which is the main problem from the numerical point of view? Wave breaking is an unsteady multiphase flow where a wide range of interfacial length scales is involved air Wave propagation O(m)! scales larger than the grid size Volume Of Fluid water air bubbles Breaking event: entrainment of air bubbles O(10 4 m)! scales smaller than the grid size method for dispersed flow
Which method to handle the motion of bubbles? Balachandar,2009: St = τ p τ ξ = ( )( 2ρ+1 36 1 1+0.15Re 0.687 p )( d ξ ) 2 ( ) 4/3 ξ η Assuming: ρ = 0.001, ξ = 0.01 m, η 1 10 5 m, d = [0.0006 0.0008 0.0010 0.0012 0.0016 0.0020 0.0025 0.0031 0.0039 0.0049 0.0062 0.0078 0.0099] m 1 0.2 Lagrangian Eulerian St [-] 0.001 Equilibrium Eulerian Dusty gas 10 0 10 1 10 2 10 3 d/η [-]
A quick look at the multiphase Eulerian methodology N-S equations discretized in a control volume with n phases inside n equations for the 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 modeled by a sub-grid term (α i ρ i u i ) t (α i ρ i ) t + (α i ρ i u i ) = S i + (α i ρ i u i u i ) = α i p + (α i ρ i T i )+α i ρ i g+m i Set of equations solved per each phase 2n equations
How can the coupling be realized? Coupling means that the VOF-like solver is combined with the multiphase Eulerian methodology. Which phases are involved? water + air above the free surface + n bubble classes = n+2! Different couplings exist, two have been tried: Eulerian framework for all phases Eulerian for bubbles in the mixture VOF for the mixture Numerical sharpening and Mi Numerical sharpening air air bubbles mixture air bubbles water
VOF with momentum transfer modeling at the free surface Glass of water Propagation of a solitary wave 0.15 Instabilities at the free surface in both cases! Surface elevation from SWL [m] 0.12 0.09 0.06 0.03 0 4 2 0 2 4 x [m] 6 8
VOF without momentum transfer modeling at the free surface Calculation of void fractions (MULES algorithm) Glass of water water (w) continuous air (air) n classes ρ mixt = α wρ w +α air ρ air α w +α air ν mixt = α wν w +α air ν air α w +α air Calculation of pressure and velocities (PISO algorithm) mixture (mixt) n classes Main achievement: modeling of the momentum transfer between water and continuous air not needed anymore!
Description of the model
Governing equations Averaged mass and momentum conservation equations: (α i ρ i u i ) t (α i ρ i ) t + (α i ρ i u i ) = S i + (α i ρ i u i u i ) = α i p + (α i ρ i T eff i )+α i ρ i g+m i The n classes have the same density and viscosity but they have different diameter S i and M i represent mass transfer and interfacial forces among phases respectively
Effective stress tensor T eff i By employing the eddy-viscosity approximation, the effective stress tensor is: [ 1 T eff = 2(νi eff ]) 2 ( u i + u T i ) 1 ] 3 ( u i)i 2 3 k ii The effective viscosity of the mixture phase ν eff is (Deen et al., 2001): ν eff = ν +ν t +ν BI ν = ν mixt and ν t is the turbulent viscosity calculated by a dynamic Smagorinsky model ν BI is an extra-term accounting for the bubble-induced turbulence (Sato et al., 1975): ν BI = 0.6 n α i d i u i u mixt i=1
Mass exchange among phases S i S i = B + i +B i +C + i +C i +E i +D i breakage Mass transfer among the n classes: - breakage B i + +Bi (Prinche et Blanch, 1990) - coalescence C i + +Ci (Martinez-Bazan et al., 1999) coalescence d j < d i < d k Mass transfer between the dispersed bubbles and air -from air into the n classes air entrainment E i -from the n classes into air degassing D i Degassing modeled as (Hänsch et al., 2012): D i = ϕ air ρ i α i / (a t t) where t is the time step, a t = 20 a constant and ϕ air = 0.5tanh[100(α air 0.5)]+0.5
Air entrainment modeling The air entrainment is reproduced by a sub-grid scale model (Derakhti and Kirby, 2014): ( ) E i = c enρ mixt 4π σ α f i mixt n ǫ i (d i sgs 2 )2 f i - c en is a parameter which regulates the amount of entrained bubbles - f i is the size spectrum of the entrained bubbles (Deane and Stokes, 2002): { ( d i f i = 2 ) 10 3 if ( d i 2 ) > 1 mm. ( d i 2 ) 3 2 if ( d i 2 ) 1 mm. - i is the width of each bubble class Bubbles are entrained at the free surface cells when ǫ sgs is larger than a fixed threshold! ǫ sgs is the rate of transfer of energy from the resolved to the sub-grid scales modeled by LES turbulence model: ǫ sgs = 2ν t 0.5( u mixt + u T mixt) 2
Momentum transfer among phases M i Between every class and mixture phase: M i = M D,i +M L,i +M VM,i +M TD,i - Drag force M D,i = 3 4 ρ mixtα mixt α i C D u i u mixt (u i u mixt ) d i - Lift force M L,i = ρ mixt α mixt α i C L (u i u mixt ) ( u mixt ) ( - Virtual mass force M VM,i = ρ mixt α mixt α i C Dumixt ) VM Dt Du i Dt - Turbulent dispersion force M TD,i = 3 4 C D ρ mixt d i (ν t +ν BI ) S b u i u mixt α i Between water and continuous air, i.e. at the free surface, the tension force is accounted as (Brackbill et al., 1992): σ = 0.0728 N/m and κ is the free surface curvature. M surf,i = σκ α (1)
Free surface sharpening method An additional term is added to the interface transport equations of water and continuous air: (α i ρ i ) t + (α i ρ i u i )+ [u c α i (1 α i )] = S i The compression velocity u c compresses the interface counteracting the numerical diffusion: u c = min(c u r,max( u r )) α α C is a coefficient that the user can specify and was taken as 1
Validation of the model
Bubble column of (Deen et al., 2001): simulation set-up z y x 1.00 m 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 d i = 4 mm - n = 11 d i = [1.0 1.2 1.6 2 2.5 3.1 4.0 5.0 6.3 8 10] mm initial water level 0.45 m measurement point (0,0,-0.20m) air 0.15 m Why this case? It resembles the motion of the bubble plume in waves. The difference is that the air entrainment is imposed by boundary conditions. 0.15 m
Bubble column of (Deen et al., 2001): results uz,w [m s 1 ] 0.3 0.2 0.1 0 experiment 1 n = 11 0.1 0.075 0.05 0.025 0 0.025 0.05 0.075 x [m] Mean water axial velocity uz,air [m s 1 ] 0.5 0.4 0.3 0.2 0.1 experiment n = 1 0 0.075 0.05 0.025 0 0.025 0.05 0.075 x [m] Mean air axial velocity αair [-] 0.05 0.04 0.03 0.02 0.01 n = 1 n = 11 0 0.075 0.05 0.025 0 0.025 0.05 0.075 x [m] Mean gas hold-up u 2 x,y,z,w [m s 1 ] 0.2 0.15 0.1 0.05 0 experiment u 2 x,w 0.05 u 2 y,w 0.1 u 2 z,w 0.075 0.05 0.025 0 0.025 0.05 0.075 x [m] Mean water turbulence fluctuations kw [m 2 s 2 ] 0.03 0.02 0.01 0 experiment 1 n = 11 0.01 0.075 0.05 0.025 0 0.025 0.05 0.075 x [m] Mean turbulent kinetic energy Spectral density [m 2 s] 10 5 10 0 10 5 10 10 u x,w u y,w u z,w 5/3 10 15 10 2 10 0 10 2 10 4 Frequency [Hz] Spectrum of turbulence 3
Bubble column of (Deen et al., 2001): the flow
Application of the model
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 the constant bottom - Breaking event by a dispersive focusing method:80 linear components of a JONSWAP spectrum (T p = 1.7 s, H s = 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 for wave generation
An isolated unsteady spilling wave: surface elevation η [m] 0.2 0.1 0 0.1 linear CFD 0.2 5 4 3 2 1 0 1 2 3 t [-] 0.2 Simulated breaking point x b 15.0 m Simulated breaking time t b 16.85 s Period of highest wave T c 1.6 s t = t t ob T c η [m] 0.1 0 0.1 linear CFD 0.2 5 4 3 2 1 0 1 2 3 t [-] Upper: x=8 m. Lower: x = x b = 15.0 m Comparison reveals second order effects
An isolated unsteady spilling wave: bubble entrainment Parameters c en and ǫ sgs of the air entrainment model need to be calibrated. How? (Lamarre and Melville, 1991) measured void fraction in similar waves c en = 20 and ǫ sgs = 0.01 m 2 s 3 V b /V 0 [ ] A b /V 0 [ ] 1.2 1 0.8 0.6 0.4 0.2 0 30 20 10 experiment 7phases 14phases 0 0.5 1 1.5 t [-] experiment 7phases 14phases 0 0 0.5 1 1.5 t [-] - Two simulations: 7 phases and 14 phases - Volume per unit length of crest: V b = α b H(α b α b thld )da A - Cross-sectional area: A b = H(α b α b thld )da A
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 (C c = L c /T c 1.8 m s 1 ) and the thickness of the roller was around 5 cm.
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
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 and acceleration - C D and C M evaluated according to Re and KC at different depths - Based on LWT: 10 Re 3 10 4 and KC 20 η [m] In line Force [N] 0.2 0.1 0 0.1 linear CFD 0.2 3 2 1 0 1 2 3 t [-] 6 4 2 0 2 4 6 Morison CFD 3 2 1 0 1 2 3 t [-]
An isolated spilling wave: impact with bubbles
An isolated spilling wave: impact without VS with bubbles Differences are recognized at the interval 0.15 t 0.4 when the bubble plume passed over the cylinder Largest differences are localized at around t = 0.25 t when the volume of the bubble plume was maximum In line Force [N] In line Force [N] 6 4 2 0 2 4 6 6 4 2 0 2 4 6 0 0.25 0.5 0.75 1 1.25 1.5 t [-] CFD no bubbles CFD with bubbles CFD no bubbles CFD with bubbles 0.1 0.15 0.2 0.25 0.3 0.35 0.4 t [-]
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 the separation - uniform current in the 3D flume - Re = 2 10 4 - C D 1.3 3 2.5 2 CD [-] 1.5 1 dynamic Smagorinsky 0.5 dynamic Smagorinsky with WF constant Smagorinsky with vd DES 0 0 10 20 30 40 50 t [s]
Conclusions
Main conclusions and on-going work A CFD simulation of the entire breaking wave process involves interfacial length scales both smaller and larger than the grid size. An Eulerian model coupled with a VOF-type interface capturing algorithm was developed to handle such multi-scale problem. A bubble column flow was analyzed to test the momentum transfer modeling and the implemented mass transfer formulations. An isolated breaking wave generated by a dispersive focusing method was simulated to test the air entrainment model The impact of the spilling wave on a slender cylinder was reproduced to estimate the effect of bubbles on the exerted in-line force. Further investigations are needed, especially concerning the flow around the cylinder.
References
References References I Balachandar S. 2009. A scaling analysis for point-particle approaches to turbulent multiphase flows. Int. J. Multiphase Flow 35:801-10 Sato, Y., and Sekoguchi, K., 1975. Liquid velocity distribution in two-phase bubble flow. International Journal of Multiphase Flow, 2(1), pp. 79-95. Martinez-Bazan, C., Montanes, J., and Lasheras, J., 1999. On the breakup of an air bubble 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 air bubble injected into a fully developed turbulent flow. Part 2. Size PDF of the resulting daughter bubbles. Journal of Fluid Mechanics, 401, pp. 183-207. Prince, M. J., and Blanch, H. W., 1990. Bubble Coalescence and Break-Up in Air-Sparged Bubble Columns. AIChE Journal, 36(10), pp. 1485-1499. Hansch, S., Lucas, D., Krepper, E., and Hohne, T., 2012. A multi-field two-fluid concept for transitions between different scales of interfacial structures. International Journal of Multiphase Flow, 47, Dec., pp. 171-182.
References References II Derakhti, M., and Kirby, J. T., 2014. Bubble entrainment and liquid-bubble interaction under unsteady breaking waves. Journal of Fluid Mechanics, 761, Nov., pp. 464-506. Deane, G. B., and Stokes, M. D., 2002. Scale dependence of bubble creation mechanisms in breaking waves.. Nature, 418(6900), Aug., pp. 839-44. Brackbill, J., Kothe, D., and Zemach, C., 1992. A continuum method for modeling surface tension. Journal of Computational Physics, 100, pp. 335-354. Deen, N. G., Solberg, T., and Hjertager, B. H., 2001. Large eddy simulation of the Gas Liquid flow in a square cross-sectioned bubble column. Chemical Engineering Science, 56, pp. 6341-6349. Rojas, G., and Loewen, M. R., 2010. Void fraction measurements beneath plunging and spilling breaking waves. J. Geophys. Res., 115(C8), p. C08001.