Hydrodynamic Analysis of Wave–induced Nonlinear Motion ofBackward Bent Buoy by the Use of the Particle Method
Mohammad Arif Hasan MAMUN, Shuichi NAGATA, Kazutaka TOYOTA,Yasutaka IMAI and Toshiaki SETOGUCHI
Institute of Ocean Energy, Saga University
1 Honjo-machi, Saga 840-8502, Japan

A fluid-structure interactions simulation program has been developed by using moving particle
semi-implicit (MPS) method for the hydrodynamic analysis of wave–induced nonlinear motion ofbackward bent buoy (BBDB). The BBDB is a floating oscillating water column type wave energy
converter. The MPS method is based on the particle interaction models and discretizes the governing
differential equation of continuum without the use of computational grid model. To satisfy theincompressibility the particle number density is implicitly required to be constant. A semi-implicit
algorithm is used for two-dimensional incompressible non-viscous flow analysis. The MPS method iscapable to simulate the complicated behavior of the free water surface. The particles whose particle
number densities are below a set point are considered as on the free surface. To compute the motionsof the BBDB, the equations of translational and rotational motions were integrated in time todetermine the correct position of the BBDB surface at each time step of the time-domain calculation.
The effect of numerical resolution on the results was checked by changing the number of particles.
Simulation is done for different amplitude and frequency of the wave generator.
Key Words : Oscillating Water Column (OWC), Wave Energy Converter (WEC), Backward Bent
Duct Buoy (BBDB), Moving particle semi-implicit (MPS) method

1. Introduction
The diminution of oil and global warming in future
encouraged a major change in the renewable energies
development and raised the interest in large-scale energy
production from the waves. Wave energy from the sea isbeing increasingly considered in many countries as amajor resource. The energy from ocean waves is apromising resource of renewable energy because of its
high energy density. The waves are produced by wind
action and are therefore an indirect form of solar energy.
Several types of wave energy converters (WEC) havebeen developed in the recent years and detail review has
been done by Falcao (1). Among these WEC the Backward
Bent Duct Buoy (BBDB) was invented under theleadership of Masuda (2) which has some advantages thanothers, i.e. primary conversion efficiency is high, a longer
floating structure is not required, and the mooring force
and mooring cost is less (3).
૒⾐ᄢቇ ᶏᵗࠛࡀ࡞ࠡ࡯⎇ⓥ࠮ࡦ࠲࡯㧔‫ޥ‬840-8502 ૒⾐⋵૒⾐Ꮢᧄᐣ↸ 1㧕
E-mail: hasan@ioes.saga-u.ac.jp

In the BBDB, the oscillating water column (OWC) ductis bent backward from the incident wave direction (Fig. 1)
which is reversed than that of the frontward facing duct
version. By making the backward bent duct, the length of
the water column could be made sufficiently large toachieve resonance, and the draught of the floating structure
could be kept within acceptable limits. The research on
BBDB converter (including model testing) has been carriedout in several countries like Japan, China, Denmark, Korea,and Ireland. The BBDB converter has been used toproduce power about one thousand navigation buoys inJapan and China (4), (5). A BBDB converter model equippedwith a horizontal axis Wells turbine has been tested in thesheltered sea waters of Galway Bay (western Ireland) since
the end of 2006 (6). A report has been prepared for theBritish Department of Trade and Industry (DTI) tocompare three types of floating OWCs for electricity
generation in an Atlantic environment: BBDB, Sloped
Buoy and Spar Buoy (7). They found that based on pure
power capture performance the BBDB is superior to theSloped buoy and Spar buoy, the comparison being for
devices with the same capture width㧚To get the optimum design of the BBDB energy

necessary. To simulate incompressibility a semi-implicitconverter some numerical investigation should be done toalgorithm has been employed. MPS method has provenestimate the motion of the floating body and mooring

useful in a wide variety of engineering applications

system, the fluctuation of air pressure in the air chamberincluding free-surface hydrodynamic flow.

etc., before starting the experiment. A number of theoreticalIn this method, the fluid is considered as assemblies ofmodels have been developed to simulate the energy

interacting particles, the motion of the particles areconversion, from wave to turbine shaft, of fixed OWC

calculated through the interactions with neighboring

plant equipped with a Wells air-turbine(8), (9). A few works

has been done on floating OWC-type wave energy
converter. Hong et al.

(10)

used the linear wave theory to

calculate the motions and time-mean horizontal drift forcesof floating backward-bent duct buoy wave energy
absorbers in regular waves taking account of the oscillating
surface-pressure due to the pressure drop in the air chamber
above the oscillating water column. Toyota et al.

(11)particles and according to the governing equations of fluidmotion, the continuity and Navier-Stokes equations:ͳ ‫ߩܦ‬൅ ‫׏‬Ǥ ࢛ ൌ Ͳሺͳሻߩ ‫ݐܦ‬‫࢛ܦ‬ͳൌ െ ‫ ݌׏‬൅ ܏ ൅ ߥ‫׏‬ଶ ࢛ሺʹሻ‫ݐܦ‬ߩ

Where, u = particle velocity vector;t = time;developed the equations of motion of a floating OWC-type

ȡ = fluid density;

wave energy converter in waves considering memoryp = particle pressure;effect by air pressure in air chamber and some calculationg = gravitational acceleration vector andresults in frequency domain by 3D boundary element

Ȟ = kinematic viscosity.

method. They have also done the experiments on exciting
forces and radiation forces and calculation results havebeen compared with experimental results in frequency
domain.

Although Eq. (1) is written in the form of a compressible
flow, but in this study, water is treated as incompressible
and the density ȡ is considered as constant. The left handside of Eq. (2) denotes the Lagrangian timedifferentiation involving the advection term and in theMPS method, this term is automatically calculated
through the tracking of particle motion.
The above equations are discretized by use ofdifferential operator models, namely the gradient andLaplacian operators. The gradient operator is modeledby using the weight function. A particle interacts with its

Figure 1: principle of BBDBThe purpose of our research is to develop a optimal
design method for the floating OWC-type WEC such as
BBDB, in order to do that as the first step, a wave
generating tank has been simulated by moving Particle
Semi-implicit method. Secondary, the exiting force byincident wave induced by the body motion and the airpressure fluctuation in chamber acting on BBDB are
obtained by using the same method.

2.

neighboring particles within some influence area
covered with a weight function w(r), where r is adistance between two particles. The weight or kernel
function used in this study is as follows:‫ݎ‬௘Ͳ ൑ ‫ ݎ‬൑ ‫ݎ‬௘െͳሺ͵ሻ
‫ݓ‬ሺ‫ݎ‬ሻ ൌ ቊ ‫ݎ‬‫ݎ‬௘ ൑ ‫ݎ‬ͲWhere,

re = radius of the influence area of each particle(kernel size).

Moving Particle Semi-implicit Method

The moving particle semi-implicit (MPS) method
has been designed to handle extremely complicatedfree-surface problems of incompressible fluids in thefield of nuclear engineering (12). In the MPS method
fluids are represented by particles and grids are not

Figure 2: Profile of weight function w(r)One important aspect of the weight function is that it isinfinity at r=0. This is good for avoiding clustering ofparticles (12).

‫݌‬Ƹ ௜ ൌ݉݅݊൫‫ ݌‬ǡ ‫ ݌‬൯ǡ‫ ܬ‬ൌ ൛݆ǣ ‫ݓ‬൫ห‫ݎ‬௝ െ ‫ݎ‬௜ ห൯ ് Ͳൟሺ͹ሻ݆‫ ܬא‬௜ ௝The Laplacian operator is formulated as [12]:When a particle i and its neighbors j are located at ri andrj, particle number density is defined as,‫ۄ݊ۃ‬௜ ൌ σ௝ஷ௜ ‫ݓ‬൫ห‫ݎ‬௝ െ ‫ݎ‬௜ ห൯ ሺͶሻ‫׏ۃ‬ଶ ‫ۄ׎‬௜ ൌWhereIt is assumed that the fluid density is approximatelyʹ‫ܦ‬௦෍൫‫׎‬௝ െ ‫׎‬௜ ൯‫ݓ‬൫ห‫ݎ‬௝ െ ‫ݎ‬௜ ห൯ ሺͺሻ
݊଴ ߣ
௝ஷ௜

σ്݆݅ ‫ݓ‬൫ห‫ ݆ݎ‬െ ‫ ݅ݎ‬ห൯ ห‫ ݆ݎ‬െ ‫ ݅ݎ‬ห

ʹconstant; this is equivalent to the particle number densityߣൌbeing constant. The constant value of the particle number

process consist of two main steps. The first step is andensity is denoted by n0.

explicit calculation to obtain intermediate or temporal

A gradient vector is estimated between two neighboring

velocities under the given viscosity and gravity terms. Butparticles i and j as

in this step, mass conservation is not satisfied; i.e. theproportional to the particle number density and forincompressible flow the fluid density is required to be൫Ȱ௝ െ ߔ௜ ൯൫‫ݎ‬௝ െ ‫ݎ‬௜ ൯ห‫ݎ‬௝ െ ‫ݎ‬௜ หଶσ്݆݅ ‫ݓ‬൫ห‫ ݆ݎ‬െ ‫ ݅ݎ‬ห൯ሺͻሻ

The MPS method is an iterative prediction-correction

number densities n* that are calculated at the end of firstprocess diverge from the constant n0. For this reason asecond corrective process is required to adjust the numberWhere, ĭ is a physical quantity. The gradient vector at ri isdensities to initial constant values prior to the time step. Ina weighted average of these vectors (Figure 3):

the second process, the intermediate particle velocities areupdated implicitly through solving the Poisson Pressure
Equation (PPE) derived as (12):ሺ‫׏‬ଶ ‫݌‬௞ାଵ ሻ௜
Where, ǻt = calculation time stepk = step of calculation.
A particle whose particle number density satisfies

Figure 3: Concept of gradient operator in standard MPS݊௜ ൏ ߚ݊଴ is regarded as the free surface. The zeropressure boundary condition is applied for the particle
The gradient operator is a local weighted average of thegradient vectors between particles i and its neighboring
particles j:‫ۄ׎׏ۃ‬௜ ൌ

which is considered as a free-surface particle. The value ofȕ can be chosen between 0.80 and 0.99 (12). We consider

ȕ = 0.97. Different sizes of the weight function have been‫׎‬௝ െ ‫׎‬௜‫ܦ‬௦෍ଶ ൫‫ݎ‬௝ െ ‫ݎ‬௜ ൯‫ݓ‬൫ห‫ݎ‬௝ െ ‫ݎ‬௜ ห൯ ሺͷሻ݊଴ห‫ ݎ‬െ ‫ ݎ‬ห௝ஷ௜௝௜used in this work. For the particle number density and thegradient model the size is re=2.1l0, where l0 is the distance
between two adjacent particles in the initial configuration.

Where Ԅ = an arbitrary scalar function,

On the other hand, for the Laplacian model the size isw(r) = the kernel function and

Before starting the simulation of BBDB, a wave

Ds = number of space dimensions,r = coordinate vector of fluid particle,

re=3.1l0.
3.

Wave Generating Tankn0 = the constant particle number density.

generating water tank (as shown in Figure 4) has been

The pressure gradient is defined by replacing Ԅi in Eq. (5)simulated by MPS method which is similar to Toyota etby the minimum value of Ԅ among the neighboring
particles, such as‫݌‬௝ െ ‫݌‬௜‫ܦ‬௦‫ۄ݌׏ۃ‬௜ ൌ ෍ଶ ൫‫ݎ‬௝ െ ‫ݎ‬௜ ൯‫ݓ‬൫ห‫ݎ‬௝ െ ‫ݎ‬௜ ห൯ ሺ͸ሻ݊଴ห‫ ݎ‬െ ‫ ݎ‬ห
௝ஷ௜

௝

௜

al. (11) experimental wave tank in size. This tank is 18 min length and 1 m in depth.

Total 9000 particles have been used to simulate the tank.

Figure 4 shows the basic BBDB model inside the wave

Reciprocating motion has been given to the left sidetank. When the solid consists of particles i, relationships

wall to generate the wave.

among co-ordinates of solid particles ri, the centre of solid
rg, relative co-ordinates of solid particles qi and the moment

yof inertia I are represented byOrifice௡

Vertical wall
0.64 mWave generatorͳ෍ ‫ܚ‬௜݊‫ܚ‬௚ ൌ‫ܙ‬௜ ൌ ‫ܚ‬௜ െ ‫ܚ‬௚

x0.38 m0.60 m௡‫ ܫ‬ൌ ෍ȁ‫ܙ‬௜ ȁଶ

1.0 m
0.85 m௜ୀଵ௜ୀଵ

0.15 mThe solid particles are calculated in each time step by using
the same procedure with the fluid particles and then

18.0 mtranslation and rotation velocity vectors of the solid are
Figure 4: Basic BBDB model inside wave generating tankcalculated by

The particles of the left side wall moved according to the௡ͳ‫ ܂‬ൌ ෍ ‫ܝ‬௜݊following equations: ݇ሺ݄ ൅ ‫ݕ‬ሻ݀‫ݔ‬ൌ ‫ ݑ‬ൌ ‫߱ܣ‬ሺ݇‫ ݔ‬െ ߱‫ݐ‬ሻሺͳͳሻ ݄݇݀‫ݐ‬Where,௡ͳ‫ ܀‬ൌ ෍ ‫ܝ‬௜ ൈ ‫ܙ‬௜‫ܫ‬u = velocity of the wave generator

௜ୀଵ

A = amplitude of the wave generator
Ȧ = angular Frequency =k = wave numberଶగ்

h = water depth௜ୀଵThen the velocity vectors of the solid particles are replacedby those of the solid motion:‫ܝ‬௜ ൌ ‫ ܂‬൅ ‫ܙ‬௜ ൈ ‫܀‬The boundary conditions are given on the free surface

x = axis in the direction of propagationand the solid surface. The free surface may be detected

y = axis in the vertically upward direction

using the particle number density only, i.e., when thet = time simulation

particle number density is below 0.97n0, the particle is

T = wave period

regarded as being on the free surface. Then a Dirichlet

The wave number, k, is determined from the following

condition for the pressure, equal to the atmosphericequation,pressure P0, is given to this particle on the free surface. On߱ଶ ݄ൌ ݄݇ ሺ݄݇ሻሺͳʹሻ݃

the solid surface, the fluid velocity is given as equal to theߟ ൌ ‫ ܣ‬ሺ݇ሺ‫ ݔ‬െ ܽሻ െ ߱‫ݐ‬ሻ ൅ ܾሺͳ͵ሻ

Two-dimensional calculation of the wave generating

The wave profile is,Where a and b are constants and ݃ is the acceleration due togravity.

4.

Simulation of BBDB

velocity of the boundary.
5.

Results and Discussions

tank and the exiting force by incident wave induced by
the body motion on BBDB has been done by moving
particle semi-implicit method with the following
parameters: distance between neighboring particles in theAt first the movement of solid particles of the BBDB is

initial configuration, l0, is 0.05m, the amplitude of thecomputed as ordinary fluid particles and then the position

wave generator is 0.2 m, wave period is 2 sec. Figure 5of each particle is corrected so as to form the shape of ashows the free surface profile, the motion of the wave

solid body in view of the conservation of momentum.

generator and BBDB at different times.

could be seen in the body motions, even for the shorter(a)at t = 8.0 sec

wave period.
References(b)

at t = 9.0 sec(c)

at t = 11.7 sec(d)

at t = 12.7 sec

Figure 5: The free surface profile, the motion of wave
generator and BBDB at different times.
The left wall particles are moving with a velocity
given by Eq. (11) to generate the waves. The right andbottom walls particles are fixed and velocities are zero.
The particles on the inner first line of the walls
participate in the pressure calculation along with otherinside water particles. The particles of the BBDB follow
the same procedure of the other water particles and thentranslate and rotate according to the calculation of afloating solid body. The water particles close to the left
wall move towards the right wall and produce the freesurface profile. Along with the free surface movement,
the BBDB is moving ups and down and the watercolumn height inside the BBDB is changing
accordingly. In this way the oscillating water column is
produced inside the BBDB which can be used toestimate power by providing generator near the orificeof the BBDB.
6.

Conclusions

Hydrodynamic analysis of wave–induced nonlinear
motion of the Backward Bent Duct Buoy (BBDB)
inside a wave generating tank has been done by themoving particle semi-implicit (MPS) method. The freesurface profile generated by the motion of the left wall
has been shown with the oscillating water column inside
the BBDB at different times. To check the numericalresolution the simulations were performed with a larger
number of particles. However, no obvious improvement(1) Falcao, A.F. de O., "Wave energy utilization: A
review of the technologies", Renewable andSustainable Energy Reviews, Vol.14, pp. 899–918,2010.
(2) Masuda Y, McCormick ME., "Experiences inpneumatic wave energy conversion in Japan",Utilization of ocean waves-wave to energy
conversion ASCE, pp. 1-33, 1986.
(3) Nagata, S., Toyota, K., Imai, I., and Setoguchi, T.,"Numerical Simulation for Evaluation of Primary
Energy Conversion of Floating OWC-type Wave
Energy Converter", Proceedings of the Nineteenth(2009) International Offshore and Polar
Engineering Conference, pp 300-307, 2009.
(4) Masuda Y, Xianguang L, Xiangfan G., "Highperformance of cylinder float backward bent ductbuoy (BBDB) and its use in European seas",
Proceedings of (First) European Wave Energy
Symposium; pp. 323–37, 1993.
(5) Masuda Y, Kimura H, Liang X, Gao X, Mogensen
RM, Andersen T., "Regarding BBDB wave power
generation plant", Proceedings of 2nd European
Wave Power Conference, pp. 69–76, 1995.
(6)Ocean
Energy.
Available
onlineat:
http://www.oceanenergy.ie.
(7) DTI. Near shore floating oscillating water column:
prototype development and evaluation. Rep URN
05/581;
2005.
Available
onlineat:
http://www.berr.gov.uk/files/file17347.pdf.
(8) Suzuki, M., Analysis of Wave Energy Converting
Characteristics of Wave Power Generating System
Installed in Breakwater, Transactions of the Japan
Society of Mechanical Engineers, Vol.70, No.700,
pp.3166-3173, 2004.
(9) Falcao, A.F. de O., Justino, P.A.P., "OWC wave
energy devices with air flow control", OceanEngineering, Vol. 26, pp. 1275–1295, 1999.
(10) Hong, D.C., Hong, S.Y. and Hong, S.W.,“Numerical Study of the Motions and Drift Force ofa Floating OWC Device,” Ocean Engineering, Vol.
30, pp 139-164, 2004.
(11) Toyota1, K., Nagata1, S., Imai, Y., and Setoguchi,T., "Research for evaluating performance ofOWC-type Wave Energy Converter “Backward
Bent Duct Buoy", Proceedings of the 8th EuropeanWave and Tidal Energy Conference, pp. 901-913,2009.
(12) Koshizuka, S. and Oka, Y., "Moving-Particle
Semi-Implicit Method for Fragmentation ofIncompressible Fluid", Nuclear Science andEngineering, Vol. 123, pp. 421-434, 1996.