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SPH for wave body interaction
Professor Peter Stansby
Osborne Reynolds Professor of Fluid Mechanics
The University of Manchester
Content
• Linear diffraction – why wave energy
• SPH introduction
• SPH weakly compressible available
• More accurate incompressible SPH in progress
• Air/water phase simulation
• Future developments.
Linear diffraction: M4 wave energy – complex problem
Efficient frequency domain
analysis using linear diffraction
• Multi –body analysis with 6 modes
• Forcing, radiation damping, added mass
from DIFFRACT (Oxford/Bath code)
• Output relative pitch, power and beam
bending moment
• Regular waves, irregular waves, multi-
directional waves – experiments in
Plymouth COAST laboratory
Papers by Liang Sun et al in J Ocean Engineering and
Marine Energy
Older configuration
Regular waves
H≈0.03m
PcREG
(W)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
T (s)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Experimental
Numerical
H≈0.05m
PcREG
(W)
0
0.2
0.4
0.6
0.8
1
1.2
T (s)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Experimental
Numerical
H≈0.05m
θr (deg)
0
2
4
6
8
10
12
T (s)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Experimental
Numerical
H ≈ 0.03 m H ≈ 0.05 m
Relative rotation
Power average
Bending moment
T (s) T (s)
Multi-directional irregular waves JONSWAP γ=1, s=5
Relative rotation
Power average
Bending moment
1.33 m/0.8 m beams and smaller floats
Plots of CWR and relative rotation: beams 1.33/0.8 m time domain model with WAMIT coefficients
CWR
θrms
Hs = 0.02m, 0.03m, 0.04m JONSWAP γ= 3.3
Multi-floats
111 121 123 132 133 134
Capture width ratio v 𝑇𝑝 for γ=3.3 from model.
Death coast
Costs based on steel mass estimates at £2000/tonne
Death Cost, Spain with scatter diagram from Iglesias and Caraballo (2009)
configuration Optimum
LSF
Average power
[kW]
Annual energy
yield [MWh]
Rated Power
[kW] Cost[M£]/MW
Total cost
[M£]
LCoE
[£/kWh]
3fl_111 120 1915 16778 5745 2.156 10.006 0.155
4fl_121 120 2948 25832 8846 1.819 12.995 0.131
5fl_131 120 3264 28596 9793 2.021 15.985 0.145
5fl_122a 120 4027 35282 12083 1.878 18.323 0.135
5fl_122b 120 3871 33914 11614 1.954 18.323 0.140
6fl_132a 120 4813 42168 14441 1.828 21.313 0.131
6fl_132b 120 4476 39210 13428 1.966 21.313 0.141
6fl_123a 120 4315 37803 12946 2.262 23.652 0.163
6fl_123b 120 4225 37017 12677 2.310 23.652 0.166
7fl_133 120 5212 45663 15638 2.110 26.641 0.152
8fl_134 120 5944 52075 17833 2.220 31.969 0.159
Leixoes
Costs based on steel mass estimates at £2000/tonne
Leixoes, Portugal with scatter diagram from Silva et al (2013)
configuration Optimum
LSF
Average power
[kW]
Annual energy
yield [MWh]
Rated Power
[kW] Cost[M£]/MW
Total cost
[M£]
LCoE
[£/kWh]
3fl_111 100 989 8665 2967 2.900 6.948 0.208
4fl_121 100 1534 13444 4604 2.427 9.024 0.174
5fl_131 100 1685 14762 5055 2.719 11.101 0.195
5fl_122a 100 2076 18191 6229 2.529 12.725 0.182
5fl_122b 100 1978 17333 5936 2.655 12.725 0.191
6fl_132a 100 2491 21829 7475 2.452 14.801 0.176
6fl_132b 100 2308 20222 6925 2.647 14.801 0.190
6fl_123a 110 2658 23285 7974 3.086 19.874 0.222
6fl_123b 110 2628 23024 7885 3.121 19.874 0.224
7fl_133 100 2646 23181 7938 2.886 18.501 0.207
8fl_134 110 3696 32385 11090 2.999 26.863 0.215
Linear diffraction observations
• Very efficient computationally
• High accuracy possible with complex problems
• Higher accuracy observed with irregular waves – possibly because reflections build up with regular waves
• Good accuracy possible with big waves
• Always good reference simulation
• BUT
Breaking effect
Luck and Benoit ICCE 2004 Regular waves
Zang et al IWWWFB 2010 Focussed waves
0.4 0.6 0.8 1 1.2 1.40.5
1
1.5
2
2.5
3
kd
FE
XP/F
M breaking
Non breaking FM is force due to max wave height Defined by Miche criterion
Stansby et al 2013 Renewable Power Generation IET
kd
Numerical wave basin needs:
• Linear diffraction reference
• Steep, breaking waves
• Wave structure dynamic interaction
• Two – phase (water/air) simulation for slam
• Computational practicality
SPH modelling Peter Stansby, Ben Rogers (SPHERIC chairman), Steve Lind, Renato
Vacondio (Parma), Alex Crespo (Vigo), Damien Violeau (EDF) with Dominique Laurence, Lin Li, Alistair Revell, Yong Wang, Lee
Cunningham Postdocs and students: Eun Sug Lee, Rui Xu, Pourya Omidvar, George
Fourtakis, Athanasios Mokos, Alex Skillen, Stephen Longshaw, Xiaohu Guo (STFC ), Antonios Xenakis, Abouzied Nasar, Alex Chow
Smoothed Particle Hydrodynamics (SPH)
• SPH is a Lagrangian particle method
• Flow variables determined according to an interpolation over discrete interpolation points (fluid particles) with kernel W
• Interpolation points flow with fluid
• Complex free-surface (including breaking wave) dynamics captured automatically
𝜙 𝑟 ≈ 𝑊 𝑟 − 𝑟′ 𝜙 𝑟′ 𝑑𝑟′ ≈ 𝑊 𝑟 − 𝑟𝑖 𝜙𝑖𝑉𝑖𝑖
Typical operators
𝜙 𝑟𝑖 = 𝑉𝑗𝜙 𝑟𝑗 𝑊 𝑟𝑖𝑗𝑗
𝛻𝜙𝑖 = −𝑉𝑗(𝜙𝑖 − 𝜙𝑗)𝛻𝑊𝑖𝑗𝑗
𝜇∆𝒖 𝑖 = 𝑚𝑗 𝜇𝑖 + 𝜇𝑗 𝒓𝑖𝑗 . 𝛻 𝑊𝑖𝑗𝜌𝑗 (𝑟
2𝑖𝑗 + 𝜂
2)𝑗
𝒖 𝑖𝑗
∆𝑝𝑖 = 2 𝑚𝑗 𝑝𝑖𝑗 𝒓𝑖𝑗 . 𝛻 𝑊𝑖𝑗𝜌𝑗 (𝑟
2𝑖𝑗 + 𝜂
2)𝑗
Basic form : weakly compressible equations
(computationally simple: no solver)
Speed of sound ~ 10 max velocity , so artificial pressure waves, noise
Stabilising options in WCSPH
• Artificial viscosity (in momentum equation)
• Shepard filter – smooths particle distribution
• XSPH – extra diffusion term in momentum eq
• δ SPH – diffusion term in continuity eq
• Shifting – purely numerical device
• Also Riemann solver formulation with artificial viscosity
3-D Numerical Wave Basin using Riemann solvers (2013)
Omidvar,P., Stansby,P.K. and Rogers,B.D. 2013 Int. J. Numer. Methods Fluids, 72, 427-452
3-D Float Simulation
t = 3.8 s
t = 4.2 s
t = 4.4 s
t = 4.6 s
VALIDATION VITAL
3-D Float Response
For a full degrees-of-freedom system, the results are very promising
Need efficient computing : DualSPHysics project for GPUs
(and CPUs)
Prof. Moncho Gómez Gesteira
Dr Alejandro J.C. Crespo
Dr José Domínguez Alonso
Dr José González-Cao
Dr Anxo Barreiro Aller
Orlando García Feal
δ SPH with artificial viscosity available in DualSPHysics
Molteni,D. and Colagrossi, A. 2009 Computer Physics Communications 180 , 861–872
Wave interaction with floating bodies , represented by particles
moving with body
Floating body subjected to a wave packet is validated with experimental data
Hadzić et al., 2015
From Alex Crespo, Vigo
Wave interaction with floating bodies
Floating moored objects Barreiro A, Domínguez JM, Crespo AJC, García-Feal O, Zabala I, Gómez-Gesteira M. Quasi-Static Mooring solver implemented in SPH. Journal of Ocean Engineering and Marine Energy, special issue
Floating moored objects Barreiro A, Domínguez JM, Crespo AJC, García-Feal O, Zabala I, Gómez-Gesteira M. Quasi-Static Mooring solver implemented in SPH. Journal of Ocean Engineering and Marine Energy, speical issue
Floating moored objects Barreiro A, Domínguez JM, Crespo AJC, García-Feal O, Zabala I, Gómez-Gesteira M. Quasi-Static Mooring solver implemented in SPH. Journal of Ocean Engineering and Marine Energy, special issue
Pelton wheel Slope
collapse
Laser cleaning
Welding
Dry laser cutting
Diverse applications for WCSPH
F1 fuel tank
sloshing
Incompressible SPH
• Noise free with numerical stabilisation (without contriving physics)
• Greater accuracy
• But requires Poisson solver for pressure so less ideal for GPUs but progress now made there
• Coupling with outer fast solvers possible
• High order possible, under development
Incompressible SPH (ISPH)
• Solves incompressible Navier-Stokes equations
𝜌𝑑𝒖
𝑑𝑡= −𝛻𝑝 + 𝜇𝛻2𝒖 + 𝒇; 𝛁 ∙ 𝒖 = 𝟎
• Incompressible SPH uses a projection method to enforce incompressibility and solves a Poisson equation for the pressure
𝛻2𝑝 =𝜌
∆𝑡𝛻 ∙ 𝒖
• Pressure field is smooth and accurate when used with particle regularisation (Lind et al., JCP, 2012, 231)
ISPH Time-Stepping Algorithm • Determine intermediate positions 𝒓𝒊
∗ = 𝒓𝒊𝑛 + ∆𝑡 𝒖𝑖
𝑛 • Determine intermediate velocity from viscous and body force terms
𝒖𝑖∗ = 𝒖𝑖
𝑛 + 𝜇𝛻2𝒖𝒊𝒏 + 𝒇𝒊 ∆𝒕/𝜌
• Pressure obtained from pressure Poisson equation for zero divergence
𝛻2𝑝𝑖𝑛+1 =
𝜌
∆𝑡𝛻 ∙ 𝒖𝒊
∗
• Intermediate velocity corrected with pressure gradient to obtain divergence-free velocity at time n+1
𝒖𝑖𝑛+1 = 𝒖𝑖
∗ − (𝛻𝑝𝑖/𝜌)∆𝑡 • Particle positions updated with centred differencing
𝒓𝑖𝑛+1 = 𝒓𝑖
𝑛 +𝒖𝑖𝑛+1 + 𝒖𝑖
𝑛 ∆𝑡
2
• Particle distributions regularised according to local particle concentration (Fick’s law, Lind et al. 2012)
𝒓𝑖𝑛+1∗ = 𝒓𝑖
𝑛+1 − 𝐷𝛻𝐶𝑖𝑛+1
• Velocities corrected using interpolation - Taylor expansion
Noise and error in 2008
Lid driven cavity Lee, E-S., Moulinec, C., Xu, R., Violeau, D., Laurence, D., Stansby, P., 2008 , JCP, 227.
WCSPH
ISPH
Accuracy and stability tests
Taylor Green vortices – 2D periodic array,
lid driven cavity,
dam breaks,
impulsive plate,
wave propagation
Above with analytical or high accuracy solutions
complex SPHERIC test cases
Taylor Green vortices – Stability Problem.
Taylor-Green vortices are simulated by ISPH_DF (Cummins
& Rudman), with 4th order Runge-Kutta time marching
scheme and random initial particle distribution.
The development of pressure field in
Taylor-Green Vortices, with ISPH_DFS,
Re=1,000
Stabilising with shifting to regularise gives highly accurate solutions
generalised Fick shifting : good for free surfaces
Concentrations become more uniform due to diffusion
Flux 𝐽 = −𝐷′𝛻𝑐
shift δ𝑟𝑠 = −𝐷 𝛻𝑐 (reduced normal to free surface)
𝑐𝑖 = 𝑉𝑗𝜔𝑖𝑗𝑗
𝛻𝑐𝑖 = 𝑉𝑗𝜔𝑖𝑗𝑗
Lind et al., JCP, 2012, 231
Dam break (wall of water problem)
Wave propagation
Non hydrostatic pressure below crest and trough
IMPULSIVE PLATE (zero gravity analytical solution from Peregrine)
Step free surface
Thin free surface layer
Skillen, et al 2013 J. CMAME, 265.
Cylinder dropping into still water
Effect thin free surface layer
Importance of air in slam force
Plate impact on wave (5.4 m/s) (represent wave impact on plate - wave on deck)
80 m/s +
Air – water coupling (ICSPH)
velocity pressure
Lind,S.J., Stansby,P.K., Rogers,B.D., Lloyd, P.M. 2015, Applied Ocean Research, 49, 57-71.
Slam of wave on a plate
Experiment (1998)
SPH domain
Pressures during slam
No air
Air
New wave in deck experiments undertaken by Qinghe Fang
Lind, Fang, Stansby, Rogers, Fourtakas ISOPE Paper No. 2017-SQY-01
2D as possible
Focussed NewWave JONSWAP
Computational SPH domains local to deck only
Boundary pressure in water from linear theory
Air velocities damped to zero in buffer zone
Focussed NewWave JONSWAP waves defined by linear theory
results
dx = 0.00125m dx = 0.025m
Results with/without air
NO AIR WITH AIR t=21.029s
t=21.039s
Approximate method for extreme inertia loading : useful fast solution
• Froude Krylov force may be accurately modelled , including breaking waves
• Added mass approximated
Taut moored buoy in COAST basin – inertia regime
Hann, M., Greaves, D., Raby, A. 2015 ‘Snatch loading of a single taut moored floating
wave energy converter due to focussed wave groups’
Ocean Engineering,2015, 96, 258–271
UKCMER
ISPH with FK forcing and empirical added mass
Lind SJ, Stansby PK, Rogers BD 2016 Fixed and moored bodies in steep and breaking waves
using SPH with the Froude Krylov approximation. J Ocean Eng Mar Energy (special issue)
Snatch loads, non breaking waves
With breaking waves snatch loads overestimated ,
initially by 30%
ISPH with FK forcing for inertia regime
• FK force accurate including breaking
• Added mass estimated – linear diffraction
• Quite accurate approximation
• Fast method especially for long crested
waves
• Potentially useful for design – same wave
may test many configurations
Future developments
Mixed Eulerian Lagrangian
Eulerian
Lagrangian
Eulerian can be high order Very high accuracy Lind SJ, Stansby PK 2016,
JCP., 326, 290–311
Future
• One method or approach challenges another
• Coupled methods – local SPH, outer potential flow – QALE-SPH-QALE near completion
• Adaptivity/variable particle size – fixed regions, or dynamic
• Eulerian – Lagrangian combined – designer CFD
• Turbulence
• Architectures – GPUs, GPU/CPU combined, cf combustion
• Numerical wave tank is close – single phase – two phase
Thanks for your attention and questions