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Numerical Study of the Effect of the Relative Depth on the OvertoppingWave Energy Converters According to Constructal Design

Elizaldo Domingues dos Santos1,a Bianca Neves Machado2,b

Marcos Moisés Zanella1,c Mateus das Neves Gomes2,d

Jeferson Avila Souza1,e Liércio André Isoldi1,f

and Luiz Alberto Oliveira Rocha3,g

1School of Engineering (EE), Federal University of Rio Grande (FURG), 96203-900 Rio Grande,Rio Grande do Sul, Brazil

2Graduate Program in Mechanical Engineering (PROMEC), Federal University of Rio Grande doSul (UFRGS), 90050-170 Porto Alegre, Rio Grande do Sul, Brazil

3Mechanical Engineering Department (DEMEC), Universidade Federal do Rio Grande do Sul(UFRGS), 90050-170 Porto Alegre, Rio Grande do Sul, Brazil

[email protected], [email protected], [email protected],[email protected], e [email protected], [email protected],

[email protected]

Keywords: constructal design, optimization, overtopping wave energy converter, numerical study,relative depth.

Abstract. The conversion of wave energy in electrical one has been increasingly studied. One

example of wave energy converter (WEC) is the overtopping device. Its main operational principle

consists of a ramp which guides the incoming waves into a reservoir raised slightly above the sea

level. The accumulated water in the reservoir flows through a low head turbine generating

electricity. In this sense, it is performed a numerical study concerned with the geometric

optimization of an overtopping WEC for various relative depths: d / λ = 0.3, 0.5 and 0.62, by means

of Constructal Design. The main purpose is to evaluate the effect of the relative depth on the design

of the ramp geometry (ratio between the ramp height and its length: H 1 / L1) as well as, investigate

the shape which leads to the highest amount of water that insides the reservoir. In the present

simulations, the conservation equations of mass, momentum and one equation for the transport of

volumetric fraction are solved with the finite volume method (FVM). To tackle with water-air

mixture, the multiphase model Volume of Fluid (VOF) is used. Results showed that the optimal

shape, ( H 1 / L1)o, has a strong dependence of the relative depth, i.e., there is no universal shape that

leads to the best performance of an overtopping device for several wave conditions.

Introduction

The global energy consumption in 2011 was approximately 1.6 × 107 MW, which is about 60%

higher than the 1980 energy consumption [1]. Moreover, the main source of energy to reach this

demand is based on the consumption of fossil fuels.

In recent years, as a result of the global energy crisis and problems associated with the use of

fossil fuels, such as greenhouse gas emissions, a great deal of research has been carried out on

renewable energies all over the world. Among these resources, the energy from the seas and oceans

has received significant attention. One important aspect is the diversity of energy forms found in

oceans: wave energy, tidal, thermal, and current energy [1-3]. Other important aspect is that the

energy from the seas and oceans has a large potential [3-6]. For example, the wave energy cancover from 15% to 66% of the total world energy consumption referred to 2006 [3-5].

Regarding the conversion of wave energy into electrical one, several operational principles have

been employed: oscillating water column (OWC), body oscillating and overtopping [6,7]. Important

Defect and Diffusion Forum Vol. 348 (2014) pp 232-244Online available since 2014/Jan/17 at www.scientific.net © (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/DDF.348.232

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reviews about the technologies, as well as, the progress around the world have been presented

[1,5,6-10].

The present work is focused on the study of the overtopping device, which consists of a ramp

that captures the water close to the wave crest and introduces it, by over spilling, into a reservoir

where it is stored at a level higher than the average free-surface level of the surrounding sea (see

Fig. 1). Potential energy of water trapped in the reservoir is then converted to electrical energythrough a low head turbine connected to a generator. Examples of installed overtopping devices are

the Wave Dragon (WD) [11] and the Sea Slot-Cone Generator (SSG) [12].

Figure 1. Sketch of an overtopping wave energy converter.

According to [13], much of the existing literature on overtopping has investigated flows over

breakwaters and dams. As the interest has been in coastal defense structures must have always high

crest freeboards and therefore low flow rates. For wave energy purposes, the maximum flow rates

are desired and low crested structures, in general, are required.

Several studies into the experimental framework have been performed for evaluation ofovertopping device parameters. For example, [11] monitored power production, wave climate,

forces in mooring lines, stresses in structure and movements in a Wave Dragon prototype of 4 MW

power production unit. In [12] were obtained experimental results to improve the knowledge about

the characteristics of a SSG overtopping WEC. The pilot plant studied was an on-shore full-scale

module in 3 levels with an expected power production of 320 MWh/year.

Into the numerical framework, [14] employed time-dependent mild-slope equations to model

single and multiple WD WECs. The results indicated that a farm of five Wave Dragon WECs

installed in a staggered grid with a distance of 2D is preferred, when taking cost and space into

account. In [15] were compared three numerical methods for estimative of the mean volume that

overcomes the ramp: Amazonia (based on the solution of non-linear equations for shallow waves),

Cobras-UC (an Eulerian model which uses VOF for the multiphase flow) and SPHysics. In [16] was

evaluated the ramp inclination of an overtopping device for a two dimensional flow. It was

considered a wave climate similar to that found in the southern part of Brazil (Rio Grande city,

placed at approximately 32ºS and 52ºW). In that work, the best geometry was reached for an angle

of 30º ( H 1 / L1 = 0.58). Afterwards, in [17] was evaluated the geometrical optimization of the ramp of

overtopping device by means of Constructal Design for a relative depth of d / λ = 0.5. In that work,

the basic idea is similar to that performed in [16], but taking into account the total volume and the

volume fraction of the ramp as constraints. In the present work, the purpose is to extend the search

for the best geometry for three different relative depths: d / λ = 0.3, 0.5 and 0.62.

It is worthy to mention that the main operational principle of other WECs has also been

numerically investigated in literature, e.g., the Oscillating Water Column was previously studied in[17-19].

Constructal theory has been used to explain deterministically the generation of shape in flow

structures of nature (river basins, lungs, atmospheric circulation, animal shapes, vascularized

Defect and Diffusion Forum Vol. 348 233



tissues, etc.) based on an evolutionary principle of flow access in time. That principle is the

Constructal law: for a flow system to persist in time (to survive), it must evolve in such way that it

provides easier and easier access to the currents that flow through it [20-24].

This same principle is used to yield new designs for electronics, fuel cells, and tree networks for

transport of people, goods and information [23-25]. The applicability of this method/law to the

physics of engineered flow systems has been widely discussed in recent literature [26-29].However, with exception of the work [18], Constructal law has not been employed yet into the

wave energy framework.

There are two main purposes of the present numerical study: the use of Constructal Design for

optimization of wave energy problems, which has been few employed in the literature and the

achievement of a theoretical recommendation about the best shape for the ramp of an overtopping

device under several wave conditions (in this specific case for various relative depths, d / λ ), which

has not been performed before at the authors knowledge. In all numerical simulations, the

conservation equations of mass, momentum and transport of volume fraction are solved by applying

the finite volume method (FVM) [29,30], more precisely the commercial code FLUENT®

[31]. The

multiphase model Volume of Fluid [32,33] is used to deal with the flow of air-water mixture and its

interaction with the overtopping device.The paper is delineated in five sections. After the introduction (Section 1), the mathematical

model, the boundary conditions of the problem, the definition of constraints and the assigned

degrees of freedom used in the optimization by means of Constructal Design are presented in

Section 2. The numerical procedures employed in the simulations are depicted in Section 3. The

results for the optimization of H 1 / L1 (the ratio between the height and the length of the ramp) which

leads to the highest mass of flow entering into reservoir for various relative depths ( d / λ ) are

obtained and discussed in Section 4. Finally, the conclusions derived from the numerical study are

presented in Section 5.

Mathematical ModelConstructal Design of Overtopping Flow. The analyzed physical problem consists of a two

dimensional overtopping device placed in a wave tank, as depicted in Fig. 2. The third dimension W

is perpendicular to the plane of the figure. The wave flow is also generated by imposing a velocity

field in the left surface of the tank. The objective of the analysis is the determination of the

optimal geometry ( H 1 / L1) that leads to the highest amount of mass of water entering in the reservoir.

The degree of freedom H 1 / L1 is optimized for various relative depths: d / λ = 0.3, 0.5 and 0.62. The

other parameters are kept fixed: ϕ = 0.02 (ratio between the ramp area and the wave tank area),

H T / LT = 0.125 and H / d = 0.23 (the ratio between the wave height and the depth). Moreover, the

following dimensions and parameters are assumed: H T = 1.0 m, LT = 8.0 m, L2 = 0.5 m, L3 = 3.0 m,

H = 0.14 m, d = 0.6 m and a total simulation time of t = 20.0 s. For d / λ = 0.3 the wavelength and the

period are given respectively by λ = 1.97 m and T = 1.15 s. For d / λ = 0.5 (λ = 1.20 m and

T = 0.88 s) and for d / λ = 0.62 (λ = 0.97 m and T = 0.8 s).

Figure 2. Computational domain of the overtopping device placed inside the wave tank.

234 Fluid Flow, Energy Transfer and Design



In the constructal design framework, the optimization of the present problem is subjected to two

constraints, namely, the total area of the wave tank constraint,

T T A H L= (1)

and the area of the ramp constraint

1 1

2r

H L A = (2)

Eq. (2) can be expressed as the device volume (area) fraction

r A

Aφ = (3)

The Multiphase Volume of Fluid (VOF). The analysis consists in finding the solution of a water-

air mixture flow. For this, the conservation equations of mass, momentum and one equation for the

transport of volumetric fraction are solved with the finite volume method (FVM).

The conservation equation of mass for an isothermal, laminar and incompressible flow with two

phases (air and water) is the following:

( ) 0vt

ρ ρ

∂+ ∇ ⋅ =

∂

r (4)

where ρ is the mixture density [kg/m³] and vr

is the velocity vector of the flow [m/s].

The conservation equation of momentum is:

( ) ( )v vv p g F t

ρ ρ τ ρ =∂

+ ∇ = −∇ + ∇ + + ∂

rr rr r (5)

where p is the pressure [N/m²], g ρ r

and F r

are buoyancy and external body forces [N/m³],

respectively, and τ is the deformation rate tensor [N/m²], which, for a Newtonian fluid, is given by:

( )2

3

T v v vI τ µ

= ∇ + ∇ − ∇ ⋅

r r r (6)

where µ is the dynamic viscosity [kg/(ms)], I is a unitary tensor and the second right-hand-side term

is concerned with the deviatory tension [N/m²].

In order to deal with the air and water mixture flow and to evaluate its interaction with the

device, the Volume of Fluid (VOF) method is employed. The VOF is a multiphase model used for

fluid flows with two or more phases. In this model, the phases are immiscible, i.e., the volume of

one phase cannot be occupied by another phase [32,33].

In the simulations of this study, two different phases are considered: air and water. Therefore, the

volume fraction concept (αq) is used to represent both phases inside one control volume. In this

model, the volume fractions are assumed to be continuous in space and time and the sum of volume

fractions, inside a control volume, is always unitary (0 ≤ αq ≤ 1). Consequently, if αwater = 0 the cell

is empty of water and full of air (αair = 1) and if the fluid has a mixture of air and water, one phase isthe complement of the other, i.e., αair = 1 – αwater. Thus, an additional transport equation for one of

the volume fractions is required:




( )( ) 0

q

qv

t

ρα ρα

∂+ ∇ ⋅ =

∂

r (7)

It is worth mentioning that the conservation equations of mass and momentum are solved for the

mixture. Therefore, it is necessary to obtain values of density and viscosity for the mixture, which

can be written by:

water w ater air a ir ρ α ρ α ρ = + (8)

and

water w ater air a ir µ α µ α µ = + (9)

Concerning wave generation, a user defined function (UDF) for the entrance velocity of the

channel is employed (see Fig. 2), simulating the wavemaker behavior [34]. The velocity

components in the wave propagation direction ( x) and in the vertical direction ( z) for the entrancechannel are based on the Stokes second order theory, and, respectively, are given by [35]:

( ) ( )

( ) ( ) ( )

( )

2

4

cosh cosh 23, , cos cos 2

sinh 4 sinh

k h z H k h z H u x y z kx t kx t

T kh TL kh

π π σ σ

+ += − + − (10)

and

( ) ( )

( ) ( ) ( )

( )

2

4

cosh cosh 23, , cos cos 2

sinh 4 sinh

k h z H k h z H w x y z kx t kx t

T kh TL kh

π π σ σ

+ += − + − (11)

where H is the wave height [m], k is the wave number given by k = 2π / λ [m-1

], d is the water depth

[m], T is the wave period [s], σ is the frequency given by σ = 2π / T [rad/s] and t is the time [s].

For other boundary conditions, the upper region of the left surface, as well as, the upper surface

have prescribed atmospheric pressure (see the dashed surfaces in Fig. 2). In the other surfaces of the

tank (lower and right surfaces) and in the device (overtopping) surfaces, the velocities are

prescribed as null. For the initial conditions, it is considered that the fluid is still and the free water

surface is d = 0.6 m.

Numerical Model

For the numerical simulation of the conservation equations of mass and momentum, a

commercial code based on the finite volume method (FVM) is employed [31]. The solver is

pressure-based and all simulations were performed by upwind and PRESTO! for spatial

discretization of momentum and pressure, respectively. The velocity-pressure coupling is performed

by the PISO method, while the GEO-RECONSTRUCTION method is employed to tackle with the

volumetric fraction. Moreover, under-relaxation factors of 0.3 and 0.7 are imposed for the

conservation equations of continuity and momentum, respectively. More details concerned with the

numerical methodology can be obtained in the works [29,30].

The numerical simulations were performed using a computer with two dual-core Intel processors

with 2.67 GHz clock and 8 GB ram memory. It is used the Message Passing Interface (MPI) for

parallelization. The time processing of each simulation was approximately 3.2 × 10

4

s (9 h).Concerning the spatial and temporal discretization, all simulations are performed with 2.0 × 105

triangular finite volumes (after the employment of grid independence study that consists of multiple

refinements) and a time-step of ∆t = 1.0 × 10-3

s.




It is important to emphasize that this numerical methodology was already validated in previous

studies of this research group [36,37]. Therefore, for the sake of brevity this validation will not be

re-presented in this study.

Results and Discussions

The optimization process consists in the simulation of several geometries with different ratios of

H 1 / L1 for each relative depth (d / λ ). To evaluate the effect of H 1 / L1 over the overtopping behavior,

Fig. 3, Fig. 4, and Fig. 5 shows the instantaneous mass flow rate of water that enters the reservoir as

a function of time for some values of H 1 / L1 and for the relative depths of d / λ = 0.3, 0.5 and 0.62,

respectively. It is worthy to mention that, for t ≥ 15.0 s the overtopping is not observed for any of

the performed simulations.

Figure 3. Instantaneous mass flow rate of water in the overtopping device as a function of time for

several values of the ratio H 1 / L1 and for d / λ = 0.3.

For the first case (d / λ = 0.3) several peaks (fluctuations of mass flow rate as function of time) are

noticed for each overtopping occurrence. This behavior is different from that obtained for d / λ = 0.5in [16]. In the latter case there is only one peak of mass flow rate for each overtopping occurrence.

Besides that, during the time interval of 4.5 s ≤ t ≤ 15.0 s only two occurrences are noticed for

d / λ = 0.5 with the highest magnitude of mass flow rate of m& = 27 kg/s for H 1 / L1 = 0.6. On the

opposite, for d / λ = 0.3, several occurrences of minor magnitudes are observed. The highest

magnitude of the mass flow rate for the latter case is m& = 15 kg/s for H 1 / L1 = 0.4. This difference in

the fluid dynamic behavior of the wave flow over the overtopping device indicates that the optimal

shape of the ramp can be affected by the wave climate. In Fig. 5 it is shown that the behavior of the

instantaneous mass flow rate as function of time for d / λ = 0.62 was similar to that reached for

d / λ = 0.5. In spite of the same behavior, the magnitudes of the overtopping peaks are significantly

suppressed for d / λ = 0.62.








In order to investigate this fact, Fig. 6 exhibits the effect of the ratio H 1 / L1 over the total amountof mass that enters the reservoir along the time for all investigated values of d / λ = 0.3, 0.5 and 0.62,

respectively.




Figure 6. The effect of H 1 / L1 on the total mass of water that enters into reservoir for various relative

depths: d / λ = 0.3, 0.5 and 0.62.

Fig. 6 shows that the extreme ratios of H 1 / L1 led to the worst performance of the overtopping

device for all relative depths evaluated. Moreover, constructal design allowed a significant increase

of the device performance. For example, for the three relative depths investigated: d / λ = 0.3, 0.5 and

0.62, the overtopped amount of water for the optimal ratio is approximately 50, 20 and 22 timeshigher than the one observed for the ratios where the lowest amount of water overtops the ramp.

For d / λ = 0.3 it is noticed only one optimal ratio of ( H 1 / L1)o = 0.4 which maximizes the mass that

enters into the reservoir along the time (mm = 9.5 kg). For d / λ = 0.5 it is observed one optimal ratio

for ( H 1 / L1)o = 0.6 which maximizes the mass of water (mm = 1.4 kg), i.e., a different optimal shape

than that found for d / λ = 0.3. Moreover, it is reached a local point of maximum for H 1 / L1 = 1.0,

which is not observed previously. For d / λ = 0.62, the maximal amount of mass that enters into the

reservoir decreases even more (mm = 0.24 kg). Other important observation is concerned with a

local point of maximum, for d / λ = 0.5 and 0.62 the presences of local peaks of total mass of water

entering the reservoir are noticed. However, the local point of maximum for d / λ = 0.62 is strongly

smoothed in comparison with the case of d / λ = 0.5. For d / λ = 0.3, it is not obtained a local point of

maximum for the total mass of water that enters into reservoir.The optimal shapes found in Fig. 6 are summarized in Fig. 7, where it is shown the effect of the

relative depth (d / λ ) over the once maximized total mass of water mm that enters into the reservoir

and over the once optimized ratio of ( H 1 / L1)o. It can be observed that there is no universal shape that

leads to the best performance for all relative depths (d / λ ) investigated. Moreover, for this specific

studied case, the once maximized total mass decreased with the increase of the relative depth. This

fact can be assigned to the higher amount of wave energy for the case d / λ = 0.3 in comparison with

the cases of d / λ = 0.5 and d / λ = 0.62. Since the waves have the same height, the wave energy is

proportional to the wave period.




Figure 7. The effect of relative depth (d / λ ) on the once maximized total mass of water that enters

into reservoir and the once optimized ratio of ( H 1 / L1)o.

Finally, to illustrate the occurrence of the overtopping, the highest instantaneous mass flow rate

of water was considered as a comparison parameter to choose one case among all studied cases.

This case is defined by φ = 0.02, H / λ = 0.12, d / λ = 0.5, H / d = 0.23, H T / LT = 0.125, H 1 / L1 = 0.6 (see

Fig. 4) and is depicted in Fig. 8. In Fig. 8 the transient flow for the mixture air-water is presentedfor the following time instants: t = 1.0 s, t = 4.0 s, t = 5.6 s, t = 8.2 s, and t = 9.2 s (Fig. 8(a) – (e)).

The initial formation of the first wave due to the imposition of the velocity fields in the inlet surface

of the wave tank is observed in Fig. 8(a). For t = 4.0 s, Fig. 8(b), the incidence of this first wave

over the ramp is noted. For t = 5.6 s, Fig. 8(c), the overtopping of the water wave flow can be

observed. While for t = 8.2 s, Fig. 8(d), the overtopping occurs again with a minor intensity. This

decrease of the amount of mass that enters into the reservoir can be concerned with the conversion

of kinetic energy into potential energy before the wave overtops the ramp, i.e., the wave reaches the

devices with lower kinetic energy. Afterwards, the flow is decreased and the wave does not go into

the reservoir again during the simulated time.

In Fig. 9 and 10 show the transient flow of the air-water mixture for the two extreme ratios of

H 1 / L1 = 0.1 and 1.2, respectively, with the purpose to illustrate the effect of the small and largeratios of H 1 / L1 over the fluid flow. The time-steps illustrated and the other parameters are the same

of the previous depicted case presented in Fig. 8. In spite of the lower height of the ramp for the

case H 1 / L1 = 0.1 (Fig. 9) the water does not spill the reservoir due to the kinetic energy dissipation

imposed by the larger length of the ramp. The behavior in this case is similar to that observed for

the flow in a beach. For H 1 / L1 = 1.2, Fig. 10 shows the conversion of kinetic into potential energies,

change which is observed in its amplitude. However, for any evaluated time steps, the increase of

the wave height is not enough to allow the overtopping of the water flow. Afterwards, the reflection

caused by the ramp induces the smoothing of the wave and the overtopping does not occur, i.e., for

this case the ramp behaves like a breakwater.




Figure 8. Occurrence of the overtopping for H 1 / L1 = 0.6 and d / λ = 0.5: (a) t = 1.0 s, (b) t = 4.0 s,

(c) t = 5.6 s, (d) t = 8.2 s, (e) t = 9.2 s.

Figure 9. Non-occurrence of overtopping for the lowest extreme ratio of H 1/ L1 = 0.1 and for

d/ λ = 0.5: (a) t = 1.0 s, (b) t = 4.0 s, (c) t = 5.6 s, (d) t = 8.2 s, (e) t = 9.2 s.




Figure 10. Non-ocorrence of overtopping for the highest extreme ratio of H 1 / L1 = 0.1 and for

d / λ = 0.5: (a) t = 1.0 s, (b) t = 4.0 s, (c) t = 5.6 s, (d) t = 8.2 s, (e) t = 9.2 s.

Concluding remarks

The present work performed a numerical study concerned with the geometric optimization of an

overtopping WEC for various relative depths: d / λ = 0.3, 0.5 and 0.62, by means of Constructal

Design. The aim was the evaluation of the influence of d / λ over the design of the ramp geometry

(ratio H 1 / L1) which leads to the highest amount of water that enters into the reservoir. The

conservation equations of mass, momentum and one equation for the transport of volumetric

fraction were solved with the finite volume method (FVM), and the Volume of Fluid (VOF) was

used to tackle with the mixture of water and air.

The results showed that constructal design led to a significantly augmentation of the device

performance for all relative depths investigated. It was also observed that the optimal shape of the

ramp, ( H 1 / L1)o, has a strong dependence of the relative depth, i.e., there was no universal shape that

conducts to the best performance of the overtopping device for all of wave conditions evaluated inthe present work. Another important observation was the presence of local points of maximum for

mass of water entering into reservoir for d / λ = 0.5 and 0.62, showing that the fluid dynamic

behavior of the wave (and the water overtopped) depends not only on the ratio H 1 / L1, but also of the

wave climate. Moreover, for all values of d / λ the worst performance was achieved for the lowest

and highest ratios of H 1 / L1.

Acknowledgements

L.A.O. Rocha thanks CNPq for the research grant. E.D. Dos Santos thanks FAPERGS for the

financial support (Process Nº: 12/1418-4). All authors thank CNPq for the financial support

(Process Nº: 555695/2010-7).




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