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