TRIPLE-FLAP WAVEMAKER BASED ON LINEAR WAVE THEORY

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol. 07, No. 01 (2015), pp. 51–63. http://jurnal.bull-math.org

TRIPLE-FLAP WAVEMAKER BASED ON

LINEAR WAVE THEORY

  

Vera Halfiani, Muarif, Marwan Ramli

_Abstract.

  This paper concerns on wave generation in hydrodynamic laboratory

deterministically. Here, it is used the linear full Laplace equation as the govern-

ing equation. The lateral boundary condition is developed based on the wavemaker

theory which focuses on triple-flap wavemaker case. The horizontal displacement

of the wavemaker is formulated and it is used to derive the potential velocity and

wave elevation. From these, the relation between stroke and wave height is found.

The result shows that both potential velocity and wave elevation are the linear su-

perposition of three monochromatic waves. Therefore, a special case which is the

trichromatic wave generation is considered. It is found that the upper and main

flap of the wavemaker contribute to form of the wave amplitude, while all flaps

produce a trichromatic wave-type.

  1. INTRODUCTION Everything related to naval architecture need a good knowledge and under- standing about the sea. For example, to build naval structures and ships, knowledge about wave characteristics is important to know accurately. Due to the complexity of wave phenomena, suddenly in the calm sea the waves were very high and steep. This wave some times called as extreme wave or giant wave. In [1] and [2] the extreme is defined as a wave whose height exceeds the significant wave height of measured wave train by factor more Received 23-11-2014, Accepted 25-01-2015.

  2010 Mathematics Subject Classification : 35C07, 35F16, 74J15 Key words and Phrases

: Triple flaps, wavemaker, extreme wave, stroke.

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  than 2.2. Occurrences of the wave are unpredictable, but their impact can cause damage to oceanic objects, i.e. ships and marine structures, that are around this wave (see Earle [3], Mori et al [4], Divinsky and Levin [5], Truslen and Dysthe [6], Smith [7], Toffoli and Bitner [8] and Waseda et al. [9]). Therefore, information of the presence of the wave is important for off- shore activities. The presence has been often reported in media. Nikolkina and Didenkulova [10] collected and analysed freak waves reported in media in 2006-2010. In order to understand the occurrence, propagation and genera- tion of the extreme wave, various studies are conducted by many researcher. Deep water observation of freak waves in the North West Pacific Ocean are conducted by Waseda et al. [11]. Hu et al. [12] studied numerically rogue wave based on non linear Schrodinger breather solutions under finite wa- ter depth. Investigation effects of dissipation on the development of rogue waves and down shifting by adding non linear and linear damping terms to the one-dimensional Dysthe equation is done by Islas and Schober [13]. Xu et al. [14] proposed (2 + 1)- dimensional Kadomtsev–Petviashvili equa- tion, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to non-linear evolution equation (NEE). Slunyaev et al. [15] observed the wave amplification in the framework of forced non linear Schrodinger equation. Ramli [16] investigated non linear evolution of wave group with three frequencies using third order approximation of Korteweg de Vries equation and maximal temporal amplitude. Ramli [17] found or- der of amplitude amplification factor of bi-chromatic wave based on third order approximation of Boussinesq equation. Ramli et al. [18] investigated third order of Korteweg de Vries equation result does not match with the experimental result by using fifth order approximation of Korteweg de Vries equation. observation of extreme wave events which are generated in the modulationally stable normal dispersion regime is done by Wabnitz et al. [19]. Peric et al. [20] regarded a prototype for spatio-temporally localized rogue waves on the ocean caused by non linear focusing and analyzed by direct numerical simulations based on two phase Navier–Stokes equations. The explicit freak waves cannot be obtained by pure intuition or by elemen- tary calculations because of itself complications is found by Blackledge [21]. Discussion about rogue waves occurring in different physical contexts and related anomalous statistics of the wave amplitude, which deviates from the Gaussian behavior that were expected for random waves can be seen in [22]. Extreme wave generation using self correcting method is studied by Fernandez et al. [23]. Xi-zeng et al. [24] simulated the extreme wave generation are carried out by using the volume of fluid (VOF) method. Ac-

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  cording to Finnegan and Goggins [25], in the design of any floating body or fixed marine structure, it is vital to test models in order to understand the fluid/structure interaction involved and inevitably laboratory experiments will be carried out in a wave tank or wave basin, followed by tests on scale models in real sea conditions. Everything near water or in the water is a subject to wave act. At beach, it will cause the movement of sand as long as the sea edge. This movement will make an erosion or destruction to the naval structures when the storm comes. In the water, off-shore oil drill platform must be capable to with- stand a great storm without damage. All the ships in the waters are target to the wave attack too, and it is uncounted how much the ship that had been sunk by the wave. That is why the study of water wave is very important in the field of marine hydrodynamics to estimate the hydrodynamic forces, motion analysis and wave pattern [26]. Water waves (surface waves) are created normally by a gravitational force in the presence of a free surface along which the pressure is constant [27]. Surface waves are the most common and important wave phenomena in the ocean for ocean engineers and naval architects [28]. According to Hermans [29], water waves are created by the presence of a free surface along which the pressure is constant. For the irrotational motion, on the free surface one than obtains the non-linear Bernoulli equation for the velocity potential function from the Euler equation. Based on small amplitude waves, lin- earised problems for the velocity potential function and for the free surface elevation are formulated. Dean and Dairymple [30] said that waves are a manifestation of forces acting on the fluid tending to deform it against the action of gravity and surface tension, which together act to maintain a level fluid surface. Ships and all off shore platform must be able to withstand all possibility caused by the waves. This is where the hydrodynamics laborato- ries take role. In these laboratories, it is possible to test the ships and other structures in model scale. It has a wave tank, which can be used to test a model with specific waves. The wave tank has a wave maker at one side and a wave absorbing that act as the beach at the other side [31]. According to ODea and Newman [32], wave makers installed in test tanks today are typically either hinged flaps (rotating about a pivot point below the still water level), or translating pistons. Furthermore they said that hinged flaps are more commonly found in deep-water tanks (water depth large compared to typical wave lengths), while pistons are more commonly found in shallow

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory tanks used to study coastal waves, Tsunamis, etc.

  In this paper, we will consider specially the flap type wave maker which is the preferred type for testing ships and structures in deep water. According to Kusumawinahyu et al. [31], deep water means water depth which exceeds approximately a third of the wavelength. As an example, the Indonesian Hy- drodynamic Laboratory (IHL) in Surabaya, East Java, Indonesia, uses single and double flap wave makers. Flap type wave makers are moving partitions which rotate around one or more horizontal axes: single flap wave makers rotate about one hinge elevation, and double flap wave makers have two degrees of freedom. Here, triple-flap wavemaker case is studied. The triple flap wave maker have three degrees of freedom.This study is an extension of the research conducted by Kusumawinahyu et al. [31], which investigate the single and double flap wavemakers in the scope of linear theory.

  2. TRIPLE-FLAP WAVEMAKER FORMULATION In this study, we consider full Laplace equation in linear form as the governing equation of water wave motion. The equation completed with its boundary conditions is express as follow

  2

  2

  ∂ φ ∂ φ

• , −h ≤ z ≤ η(x, t), x ≥ s(z, t) (1)

  2

  2

  ∂x ∂z with boundary conditions: o The bottom boundary condition:

  ∂φ = 0, z = −h (2)

  ∂x o The free surface kinematic boundary condition: ∂η ∂φ

  = , z = 0 (3) ∂t ∂x o The free surface dynamic boundary condition:

  1 ∂φ η + , z = 0 (4) g ∂t Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  o lateral boundary condition at the wavemaker: ∂φ ∂s(z, t)

  = , x = 0 (5) ∂x ∂t where φ is the velocity potential, η is the elevation above the still water level, x is horizontal space dimension, z is vertical space dimension, t is time dimension, h is the water depth, and s(z, t) represents the horizontal displacement of the wavemaker, and g is gravitational acceleration.

  The lateral boundary condition with single flap type wavemaker had been explained in [30] and [31], while the one with double-flap wavemaker had been thoroughly described in [31]. Here, it will be analysed the linear theory of water wave motion for triple-flap wavemaker case.

  Figure 1: Triple-flap wavemaker structure. Firstly, it will be derived the formula for s(z, t) for triple-flap wavemaker. Figure 1 presents the structure of the wavemaker at one side of a water basin while the other side is the wave absorber. Let h be the water depth in the tank, d , d , and d be the distance of hinges of upper flap, middle flap, and

  1

  2

  3

  main flap respectively from the still-water level. Suppose that the upper flap moves with frequency ω

  1 and has a maximum stroke S 1 , the middle

  flap moves with frequency ω and has a maximum stroke S ,and the main

  2

  2

  flap moves with frequency ω

  3 and has a maximum stroke S 3 .This scheme is illustrated in Figure 2.

  The flaps motion is divided into three major cases; only one of the flaps moves, only two of the flaps move, and all flaps move. The first case is just the same as the single-flap wavemaker theory, whichever the flap that

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory Figure 2: Scheme of triple-flap wavemaker motion.

  moves. The second case is also similar to the double-flap wavemaker case, no matter which two of the flaps that move. Assume that the all flaps move case happens. Let z be any point on the z-axis line below the still-water level (z = 0) and consider the main flaps movement. As the main flap moves, the horizontal position of water near the flaps at those points changes. Let s (z) be the distance of the position

  3

  change of water from the equilibrium state at any point z caused by the main flap. Applying the concept of similar triangles, it is obtained the following relation. d + z

  1

  

3

  s (z) = S . (6)

  3

  3

  d

  2

  3 Since the flaps oscillates with frequency 3 every time, the horizontal position also changes over time. Therefore, it can be written as follow.

  d

  1

  3 + z

  s (z) = S sin(ω t). (7)

  3

  3

  3

  d

  2

  2

  1

  water is also affected by the middle flap. By using the same idea with the main flaps movement, the horizontal displacement caused by middle flap can

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  produced by the main and middle flap. On the interval −d

  1 ≤ z ≤ 0, the

  three of the flaps cause the displacement, the similar idea as the previous is applied. In summary, suppose that all flaps move, the horizontal displacement s(z, t) is given by _{  z z z 1 1 2 2 h“ ” “ ” i h“ ” “ ” “ ” i 1 + S 1 + S d2 d3 d1 d2 d3 z z 1 sin(ω 2 sin(ω 2 t ) + 1 + S 1 t ) + 1 + S 3 sin(ω 2 sin(ω 3 t ) , −d 2 t ) + 1 + S 3 sin(ω 3 t ) , −d 2 ≤ z < −d 1 ≤ z ≤ 1} s(z, t) = h“ ” i _{ 1 2 1 + d3 z S t , −d ≤ z < −d 3 sin(ω 3 ) , −h ≤ z < −d 3 3 2}

  (8)

  3. WAVE HEIGTH - STROKE RELATIONSHIP The solution of full Laplace equation (1) - (5) can be determined us- ing the method of separation of variables. In [30] and [31], its solution using a single-flap wavemaker lateral boundary condition had been found. Kusumawinahyu et al. [31] had developed this theory to more advance level using double-flap wavemaker case. Suppose that the resulting wave mo- tion is formed by linear superposition of waves produced by each flap, the velocity potential of the triple-flap wavemaker condition is stated as follow. g cosh k (z + h)

  1

  φ(x, z, t) = A

  1 sin(k 1 x − ω 1 t)

  ω cosh k h

  1 _{" # ∞}

  1 _X

_[n]

_x

[n]

  g [n] −κ cos κ (z + h)

  1 C e cos(ω t) ¹

• 1

  1

  ω

  1 cos κ _n 1 h =1

  g cosh k (z + h)

  2

  2

  2

  ω cosh k h

  2 _{" # ∞}

  

2

_X

_[n]

[n]

  g −κ x cos κ (z + h)

  [n]

  2 C e cos(ω t) ²

• 2

  2

  ω cos κ h

  2 _n

  2 =1

  g cosh k (z + h)

  3

• A sin(k x − ω t)

  3

  3

  3

  ω

  3 cosh k _{" # ∞}

3 h

_X

_[n]

[n]

  g x cos κ (z + h)

  [n] −κ ₃

  3 C +

  e cos(ω

  3 t) (9)

  3

  ω cos κ h

  3 _n

  3 =1

  The first, third, and fifth terms of the right hand side of equation (9) are

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  second, forth, and sixth terms are associated with standing waves which will decaying over space and are often called as evanescent mode. Parameter k and κ are the wave number of progressive wave and evanescent mode respectively, which satisfy the dispersion relation _i

  2 _{i i i i}

  ω = gk tanh k h = −gκ tan κ h (10) _i [n] for i = 1, 2, 3. The coefficient A and C i are to be determined. By applying the lateral boundary condition (5), it is obtained: _{i i i i i i i i} sinh k h cosh k (h − d ) + k d sinh k h − cosh k h _i

  A = 2 _{i i i i} S (11) k d 2k h + sinh 2k h and _!

  [n] [n] [n] [n] [n] _{i i}

  sin κ i h cos κ i (h − d ) − κ i d sin κ i h − cos κ i h

  [n] _i

  C i = −2 S (12)

  [n] [n] [n] _i

  κ d 2κ h + sin 2κ h _{i i i} for i = 1, 2, 3. The surface elevation is derived from the free-surface dynamic bound- ary condition (4) at still-water level z = 0.

  1 ∂φ _z η(x, t) = − |

  =0

  g ∂t _{X ∞ [n] x}

  [n] −κ ₁

  η(x, t) =A cos(k x − ω t) + C e sin(ω t)

  1

  1

  1

  1 _n

  1 =1 _{X ∞ [n] x} [n] −κ ₂

• A cos(k x − ω t) + C e sin(ω t)

  2

  2

  

2

  2 _n

  2 _{X ∞} =1 _{−κ [n] x} [n] ₃

• A cos(k x − ω t) + C e sin(ω t) (13)

  3

  3

  

3

  3 _n

  3 =1

  The wave height the resulting wave is determined where the wave is far from the wavemaker. At this position, the evanescent mode vanish. The value of wave height H is equal to two time the amplitude A, so it can be written as A = H/2. Therefore, the elevation of wave at the position far from the wavemaker is Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  H

  1 H

  2 H

  3

  η(x, t) = cos(k x − ω t) + cos(k x − ω t) + cos(k x − ω t) (14)

  1

  1

  2

  2

  3

  3

  2

  2

  2 where H , H , and H are directly related with maximum stroke S ,

  1

  2

  3

  1 S 2 , and S 3 respectively. This relation can be determined through equation

  (11). The wave height stroke relations are given by sinh k h cosh k (h − d ) + k d sinh k h − cosh k h

  1

  1

  

1

  1

  1

  1

  1 H 1 = 4

  S

  1 (15)

  k d 2k h + sinh 2k h

  1

  1

  

1

  1

  sinh k h cosh k (h − d ) + k d sinh k h − cosh k h

  2

  2

  

2

  2

  2

  2

  2 H = 4

  S (16)

  2

  2

  k d 2k h + sinh 2k h

  2

  2

  

2

  2

  sinh k h cosh k (h − d ) + k d sinh k h − cosh k h

  3

  3

  

3

  3

  3

  3

  3 H = 4

  S (17)

  3

  3

  k d 2k h + sinh 2k h

  3

  3

  

3

  3

  4. WAVE HEIGTH - STROKE RELATIONSHIP Equation (14) shows that the outcome wave of triple-flap wavemaker is the superposition of three monochromatic waves with sufficiently different frequencies. Suppose that a trichromatic wave is generated. This wave can be considered as that a monochromatic wave having amplitude a and frequency σ is being disturbed by a couple of monochromatic waves having lower amplitude ǫa and different frequencies σ + ν and σ − ν where ν is the interval of perturbing frequency. It will be inspected that how the frequencies of flaps affect the frequencies of this type of wave.

  Figure 3: Spectrum of trichromatic wave, frequency versus amplitude. In the analysis of frequency range, Kusumawinahyu et al. [31] stated that

  Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

  main flap’s frequency. Following this theory, assume that in the triple-flap wavemaker case, the upper flap has higher frequency than middle flap, and the middle flap has higher frequency than main flap. Therefore, ω = σ + ν,

  1

  ω = σ, and ω = σ − ν, with corresponding wave number k = γ + τ ,

  2

  3

  1

  k = γ, and k = γ − τ and amplitude A = ǫa, A = a, and A = ǫa can

  2

  1

  

1

  2

  3

  be assigned. Then, the elevation can be set as: η(x, t) = ǫa cos(k x − ω t) + a cos(k x − ω t) + ǫa cos(k x − ω t)

  1

  1

  2

  2

  3

  3

  = ǫa cos((γ + τ )x − (σ + ν)t) + a cos(γx − νt) + ǫa cos((γ − τ )x − (σ − ν)t) = a(2ǫcos(τ x − νt) + 1)cos(γx − σt) (18)

  Equation (18) shows that the wave has amplitude in the form of a(2ǫ cos(τ x− νt) + 1) which is also called as modulus. This amplitude is formed by the contributions from upper flap and main flap, since ν = (ω − ω )/2 and its

  1

  3 corresponding wave number τ = (k − k )/2.

  1

  

3

  5. CONCLUDING REMARKS It has been shown that the wave produced by triple-flap wavemaker can be considered as linear superposition of monochromatic waves. These monochro- matic waves are the wave that corresponds to each flap. The amount of wave height depends on the stroke introduced by the flaps. In the trichromatic wave signalling problem, the amplitude-pattern of the wave is affected by the main and upper flap since the frequency of the amplitude pattern is formed from combination of the main and upper flap’s frequency, while the three flaps create a trichromatic wave type.

  ACKNOWLEDGMENT

  This research is funded by Syiah Kuala University, Ministry of Edu- cation and Culture Republic of Indonesia through Competency Research, Contract Number : 499/UN11/S/LK-BOPT/2014 dated 26 May 20145.

  

Vera Halfiani et. al. – Triple-Flap Wavemaker Based On Linear Wave Theory

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  Vera Halfiani, Muarif : Mathematics Graduate Study Program, Dynamic Appli-

cation and Optimization Group, Syiah Kuala University, Postal Code 23111, Banda

  Aceh, Indonesia.

  Marwan Ramli* : Department of Mathematics, Dynamic Application and Opti-

mization Group, Syiah Kuala University, Postal Code 23111, Banda Aceh, Indone-

sia.

• E-mail: marwan.math@unsyiah.ac.id or ramlimarwan@gmail.com