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Eﬀect of dual vertical porous baﬄes on sloshing reduction in a swaying rectangular tank

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⁎

I.H. Choa, , M.H. Kimb a b

Department of Ocean System Engineering, Jeju National University, Jeju 690-756, Republic of Korea Department of Ocean Engineering, Texas A & M University, United States

A R T I C L E I N F O

A BS T RAC T

Keywords: Dual vertical porous baﬄes Sloshing Matched eigenfunction expansion method Experiment MPS (moving particle semi-implicit) Resonant frequency

Liquid sloshing inside tanks of a vessel may result in increased/decreased vessel motions or structural damages. The resonant sloshing motions can be suppressed by using baﬄes inside a tank. Especially, more energy dissipation is possible by using porous baﬄes. Here, the eﬀect of dual vertical porous baﬄes on the sloshing reduction inside a rectangular tank is investigated both theoretically and experimentally. The matched eigenfunction expansion method is applied to obtain the analytic solutions in the context of linear potential theory with porous boundary conditions. The porosity eﬀect is included through inertial and quadratic-drag terms. The theoretical prediction is then compared with a series of experiments conducted by authors with harmonically oscillated rectangular tank at various frequencies and baﬄe parameters. The measured data reasonably correlate with the predicted values. It is found that the dual vertical porous baﬄes can signiﬁcantly suppress sloshing motions when properly designed by selecting optimal porosity, submergence depth, and installation position.

1. Introduction A partially ﬁlled liquid tank in a ship may experience violent liquid motions called sloshing when the ship motion frequencies are close to the sloshing natural frequencies. This violent liquid motion has gained a lot of attention in coastal and oﬀshore engineering since it is directly related to the possibility of damage on the tank wall and may aﬀect the safety of waterway transportation and oﬄoading. In this sense, many studies regarding sloshing have focused on suppression methods to minimize sloshing induced loads. Among the sloshing-suppression devices, various types of inner baﬄes have been proposed. The inner baﬄes are generally used as passive sloshing-damping devices in the liquid tank to obstruct some of the lowest resonant-mode motions. The basic concept of the passive sloshing damper is to obstruct the sloshing-induced ﬂow, dissipate the sloshing energy by viscosity, and change the lowest sloshing natural frequency to lower value. Earlier attempts to make accurate predictions of the sloshinginduced dynamic pressures on the inner baﬄes and tank walls were made by Abramson (1966). Later, an analytical study on the eﬀects of a vertical baﬄe on the resonant frequencies of ﬂuid in a rectangular tank was performed by Evans and McIver (1987). They observed that surface-piercing baﬄe changed the sloshing resonant frequencies signiﬁcantly, whereas the eﬀect of a bottom-mounted baﬄe was negligible. Other researchers also observed that vertical baﬄes not

⁎

Corresponding author. E-mail address: [email protected] (I.H. Cho).

http://dx.doi.org/10.1016/j.oceaneng.2016.09.004 Received 3 June 2016; Received in revised form 4 August 2016; Accepted 6 September 2016 0029-8018/ © 2016 Elsevier Ltd. All rights reserved.

only remarkably changed the sloshing natural frequencies but also reduced the sloshing amplitudes and induced loads on tank walls (Armenio and Rocca, 1996; Wu et al., 2013; Jung et al., 2012). In addition to vertical baﬄes, alternative shapes of inner baﬄes have been incorporated into tanks, such as annular baﬄes and ﬂexible baﬄes in cylindrical tanks (Biswal et al., 2004), horizontal baﬄes in cubic tanks (Akyildiz and Unal, 2005; 2006; Liu and Lin, 2009), and annular baﬄes in rectangular tanks (Panigrahy et al., 2009). The forces acting on baﬄes due to resonant sloshing are typically large and they can be signiﬁcantly reduced by using porous inner baﬄes instead of solid ones. Compared to solid baﬄes, porous baﬄes signiﬁcantly damp out the sloshing in a moving tank, which results in reduced forces on them and tank walls and smaller vessel motions. Tait et al. (2005) investigated a tuned liquid damper (TLD) equipped with vertical perforated screens under 2D excitation. Their experimental results showed that perforated screens worked well as TLDs. Dodge (2000) experimentally demonstrated that the porosity of vertical baﬄes reduced the slosh dynamics and the change in its natural frequency required a porosity of 10% or less. Recently, Faltinsen et al. (2011a,b,c) and Faltinsen and Timokha (2011) conducted detailed studies on liquid motions in a rectangular tank with a vertical slat-type screen in the middle. They observed that the resonance frequencies of the tank with a porous screen were diﬀerent from those of the baﬄe-free tank and the resonant sloshing

Ocean Engineering 126 (2016) 364–373

I.H. Cho, M.H. Kim

∇2 Φ = 0

frequencies depended on the solidity ratio(unity minus the porosity), submerged screen gaps, liquid depth, and the position of the perforated openings relative to the mean free surface. Cassolato et al. (2010) experimentally studied a TLD with inclined slat screens and observed that the energy loss coeﬃcient of a screen decreased with the increase in the angle of inclination. Many other theoretical and experimental researches have also been conducted with regard to the wave/sloshing energy dissipation by porous plates. Most researchers (Yu, 1995; Chwang and Wu, 1994; Wu et al., 1998; Cho and Kim, 2008; Liu et al., 2008) applied Darcy's law as the boundary condition for ﬂuids across porous plates. Darcy's law suggested that the velocity diﬀerence across the plate with relatively ﬁne pores can be related to the pressure drop by using a complexvalued frequency dependent parameter, which accounts for both viscous and inertial eﬀects. Crowley and Porter (2012a) proposed a linearized model for the wave energy dissipation through thin vertical porous barrier, where both the inertial and quadratic drag eﬀects were included. The model was then applied to the sloshing problems in a 2-D rectangular tank with a vertical screen in the middle with its bottom fully extended to tank bottom. Crowley and Porter (2012b) extended the analytic solutions of the single porous screen to the case of N vertical porous screens fully extended to tank bottom. Molin and Remy (2013) also carried out the experimental and numerical study of the sloshing motion in a rectangular tank with single perforated screen, which is again fully extended to tank bottom. The main diﬀerence between Crowley and Porter (2012a) and Molin and Remy (2013) is that Crowley and Porter (2012a) used the depth-averaged drag coeﬃcient, whereas, in Molin and Remy’s formulation, the drag coeﬃcient was varied with baﬄe’s depth. Hyeon and Cho (2015) experimentally demonstrated that the porosity of the vertical baﬄe inﬂuences both the wave elevations and dynamic pressures on tank wall. In the present paper, we obtained the analytic solutions for the liquid sloshing with dual vertical porous baﬄes of arbitrary submergence depths in a sway-oscillated rectangular tank following the methodology similar to Crowley and Porter (2012a) and Molin and Remy (2013). As far as authors know, the formulations for partially submerged dual porous baﬄes in a sway-oscillated rectangular tank are new in the open literature. In addition, a series of experiments were conducted in the 2D rectangular tank with dual vertical porous baﬄes to validate the derived analytical solutions. The dual vertical baﬄes were each located at 1/3 and 2/3 of tank's length, which are close to the two antinodes of the 2nd anti-symmetric sloshing mode. The eﬀects of various baﬄe positions were also numerically investigated. The viscous eﬀects in limiting resonant motions were also observed by using MPS (moving particle semi-implicit) computation, grid-less Navier–Stokes solver. The eﬀects of the porosity and the position/submergence depth of baﬄes on sloshing motions are systematically investigated. After checking the reliable correlation between the predicted and measured values, the analytic solutions are used to ﬁnd the optimal design of baﬄes through an extensive parametric study. Finally, main conclusions of this study are stated.

(1)

with the following boundary conditions

g

∂Φ + Φtt = 0, ∂z

∂Φ = 0, ∂z

on z = 0

on z = − h

Φx = ωξ cos ωt ,

(2) (3)

on x = ± a

(4)

where g is the gravitational acceleration. The additional boundary conditions are required to relate the ﬂows on both sides of porous baﬄe. The approximate boundary conditions at porous baﬄe were derived by Bennett et al. (1992) and Mei et al. (1974). +

⎡ ∂Φ(x, z, t ) ⎤ x ⎢ ⎥ = 0, ⎣ ⎦ x− ∂x

on x = aj , − h ≤ z ≤ − dj

(5)

∂Ur (z, t ) ⎧α ⎪ U (z , t ) U (z , t ) + 2C ⎡ ∂Φ(x, z, t ) ⎤ x 0, on −dj ≤ z ≤ 0 r ∂t ⎨2 r ⎢ ⎥ =⎪ ⎣ ⎦ x− ⎩ 0, ∂t on −h ≤ z ≤ − dj +

(6) where the square brackets denote the jump in the enclosed quantity. Ur (z, t ) = Φx − ωξ cos ωt is the horizontal velocity of the ﬂuid relative to that of the swaying tank and α and C are empirically determined coeﬃcients. α is the drag coeﬃcient related to the energy dissipation across the porous baﬄe. C represents an inertial (or blockage) coeﬃcient accounting for the added inertia felt by the ﬂuid as it accelerates. It can be neglected when the baﬄe is thin and the size of holes is not large. Mei (1989) suggested the drag coeﬃcient α with sharp edged oriﬁce Cc

⎛ 1 ⎞2 − 1⎟ α=⎜ ⎝ PCc ⎠

(7)

where P is the porosity of baﬄe. The empirical form of Cc is given by Tait et al. (2005)

Cc = 0.405e π (P −1) + 0.595

(8)

Assuming harmonic motion of frequency ω , the velocity potential, wave elevation, and horizontal relative velocity at porous baﬄe can be written as

Φ(x, z, t ) = Re{ωξϕ(x, z )e−iωt }, ζ (x, t ) = Re{η(x ) e−iωt }, Ur (z, t ) = Re{ur (z )e−iωt }

(9)

The boundary value problem Eqs. (1)–(4) can be rewritten with the velocity potential ϕ as follows:

⎧∇2 ϕ = 0, ⎪ ⎪ ∂ϕ − ω2 ϕ = 0, on z = 0 ⎪ ∂z g ⎨ ⎪ ∂ϕ = 0, on z = − h ⎪ ∂z ⎪ ∂ϕ = 1, on x = ± a ⎩ ∂x

2. Mathematical formulation and analytic solutions We investigate the two dimensional sloshing motion in a rectangular tank of the length 2a with dual vertical porous baﬄes. Cartesian axes are chosen with the x -axis along the mean free surface and z -axis pointing vertically upwards. The water depth is denoted by h , the position of dual baﬄes by x = aj , (j = 1, 2), and the submergence depth of corresponding baﬄes by dj , (j = 1, 2). Tank is forced to oscillate horizontally with amplitude ξ and frequency ω. It is assumed that the ﬂuid is incompressible and inviscid, and the wave motions are small so that linear potential theory can be used. The ﬂuid particle velocity can then be described by the gradient of a velocity potential Φ(x, z, t ), which satisfy the Laplace equation

(10)

By means of matched eigenfunction expansion method (MEEM), the ﬂuid domain is divided into three regions (I), (II), (III), as shown in Fig. 1. The velocity potential in each ﬂuid region satisfying the twodimensional Laplace equation and boundary conditions, can be written as follows: ∞

ϕ(j ) =

∑ n =0

1 (j ) −kn(x − aj −1) [An e + Bn(j )e kn(x − aj −1)]fn (z ), (j = 1, 2, 3) kn

(11)

For convenience, we include the tank walls by deﬁning a 0 = − a and 365

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across the porous baﬄe. The quadratic velocity dependence in the boundary condition (6) implies that a sway tank motion with a single frequency ω introduces multi-frequency responses of ﬂuid. Thus, expanding Φ in a Fourier time-series in multiples of ω and retaining just the fundamental frequency response (the Fourier coeﬃcient of ﬁrst term is 8/3π whilst the next term at frequency 3ω is much smaller than ω and justiﬁes this assumption) transforms Eq. (6), using (9), into following porous boundary condition with C = 0

⎧ iβj(z ) ⎪ (u (z ) − 1), on −dj ≤ z ≤ 0 ϕ(j +1)(aj , z ) − ϕ(j )(aj , z ) = ⎨ ω j (j = 1, 2) ⎪ 0, on −h ≤ z ≤ − dj ⎩ (18) 4αjξω

denotes the modulus of the complex where βj (z ) = 3π uj (z ) − 1 , number. Upon substituting Eq. (17) into Eq. (18), the following equation can be obtained:

Fig. 1. Deﬁnition sketch of a rectangular tank with dual vertical porous baﬄes.

a3 = a . The eigenvalues (k 0 = − ik , kn, n = 1, 2, ...) satisfy the dispersion relation kn tan knh = − ω2 / g and the normalized eigenfunctions can be written as

fn (z ) = Nn−1 cos kn(z + h ), (Nn )2 =

1⎛ ⎜1 2⎝

+

⎧ ⎞ ⎛ γn ∞ 1 1 ∑n =0 ⎨− k sinh( + ⎜ k tanh k s + k tanh k s ⎟u1, n − kns1) n1 n n 2⎠ ⎝ n ⎩ n iβ ⎧ ∞ ⎪− 1 ∑ (u − γn )fn (z ), −d1 ≤ z ≤ 0 =⎨ ω n =0 1n ⎪ 0, −h ≤ z ≤ − d1 ⎩

n = 0, 1, 2, ....

sin 2knh ⎞ ⎟. 2knh ⎠

⎫ ⎧ ⎞ ⎛ γ u1n ∞ 1 1 ∑n =0 ⎨− k sinh + ⎜ k tanh k s + k tanh k s ⎟u 2n − k sinhn k s ⎬fn (z ) k s n 2 n 2 n n 3⎠ n n 3⎭ ⎝ n ⎩ n iβ2 ⎧ ∞ ⎪− ∑ (u − γn )fn (z ), − d 2 ≤ z ≤ 0 ⎨ ω n =0 2n =⎪ 0, − h ≤ z ≤ − d2 ⎩

0

∫−h fm (z)fn (z)dz = δmn,

(13)

where δmn is the Kronecker delta function deﬁned by δmn = 1 if m = n , and δmn = 0 if m ≠ n . From the boundary conditions (ϕx(1)(a 0 , z ) = ϕx(3)(a3, z ) = 1) on tank walls, the following equations can be obtained

Bn(3) = γne−kns3 + An(3) e−2kns3

(14)

where sj = aj − aj−1, (j = 1, 2, 3) and γn =

1 h

⎛ 1 ⎜ k tanh k s + m1 ⎝ m

0

∫−h fn (z )dz .

The unknown coeﬃcients An(j ) , Bn(j ), (j = 1, 2, 3) in Eq. (11) can be determined by invoking the boundary conditions given by Eqs. (5) and (6) at x = aj , (j = 1, 2). The continuity of horizontal ﬂuid velocity at x = aj , (j = 1, 2) requires that ∞

ϕx(j )(aj , z ) = ϕx(j +1)(aj , z ) = uj (z ) =

∑ ujnfn (z ) n =0

(15)

If substituting Eq. (11) into the equation Eq. (15) and using Eq. (14), the unknowns An(j ) , Bn(j ), (j = 1, 2, 3) can be expressed by u1n , u 2n as follows:

=

An(2) = An(3)

=

u1n − γne kns1

, Bn(1) e kns1 − e−kns1

e kns1 − e−kns1

u2n − u1ne kns2

u2n − u1ne−kns2

(2) −kns2 , Bn =

e kns2 − e

γn − u2ne kns3 e kns3 − e−kns3

,

Bn(3)

e kns2 − e−kns2 e kns3 − e−kns3

(16)

Using Eq. (16), the velocity potential in each region can be rewritten as ∞

ϕ(1) = ∑n =0 ∞

ϕ(2) = ∑n =0 (3)

ϕ

=

fn (z ) kn sinh(kns1)

[u1n cosh kn(x − a 0 ) − γn cosh kn(x − a1)]

fn (z )

u kn sinh(kns2) [ 2n

fn (z ) ∞ ∑n =0 k sinh( [γ kns3) n n

cosh kn(x − a1) − u1n cosh kn(x − a 2 )]

cosh kn(x − a 2 ) − u 2n cosh kn(x − a3)]

iβ1(l −3/2)

=k

γm m tanh k ms1

+

−k

l) u1(m m sinh k ms2

⎛ 1 + ⎜ k tanh k s + m 2 ⎝ m

=k

γm m tanh k ms3

+

ω

iβ2(l −3/2) ω

+

iβ1(l −3/2) ω

u2(lm)

N

(1) (l ) ∑n =0 Gmn u1n −

k m sinh k ms2

N

(1) ∑n =0 Gmn γn

(20a)

⎞ (l ) 1 ⎟u k m tanh k ms3 ⎠ 2m

+

iβ2(l −3/2) ω

N

(2) (l ) ∑n =0 Gmn u 2n

N

(2) ∑n =0 Gmn γn

(20b)

0

(j ) Gmn =

1 h

βj(l )

4αjξω

=

∫−d fn (z )fm (z )dz, j

3π

N

∑n =0

uj(,ln) − γn dj

where the superscript

0

∫−d fn (z )dz ,

(j = 1, 2)

j

(l ) in Eq. (20a,b) means the present iteration step. In above equations, iteration step (l − 3/2 ) means that the averaged values between the previous two iteration steps (l − 2 ) and (l − 1) are used to compute βj . In this way, relaxation is introduced in the iterative scheme and convergence is faster. Convergence to give a desired accuracy (l +1) (l ) ujn − ujn ≤ 10−6 is reached within 10–20 iterations with the coeﬃ(1) cients ujn , (j = 1, 2, n = 0, 1, 2, .. , N ) initially taken equal to zero. By solving the above coupled algebraic Eq. (20a,b), the unknown constants u1n , u 2n(n = 0, 1, 2, .. , N ) can be determined. Subsequently, complete solutions in each region can be obtained from Eq. (17). The sloshing motion of the ﬂuid exerts hydrodynamic forces on the tank wall expressed as Fw = Re{fw e−iωt }, and these can be found analytically by integrating dynamic pressure P(z, t ) = Re{p(z ) e−iωt }over the tank walls.

γn − u2ne−kns3

=

⎞ (l ) 1 ⎟u k m tanh k ms2 ⎠ 1m

where

u1n − γne−kns1

=

(19b)

Multiplying both sides of Eq. (19a,b) by {fm (z ): m = 0, 1, 2, ... }and integrating with respect to z from −h to 0, we obtain the coupled nonlinear algebraic equations taking the ﬁrst N terms in the inﬁnite series. In this study, the iterative scheme is used for solving the nonlinear equations

Bn(1) = γn + An(1)

An(1)

(19a)

(12)

The eigenfunctions fn (z ) are complete on the interval [ − h , 0], therefore satisfy the following orthogonal relation

1 h

⎫

u2n ⎬f (z ) kn sinh kns2 n ⎭

(17)

The unknown coeﬃcients u1n , u 2n can be determined by using the remaining boundary condition representing the energy dissipation 366

Ocean Engineering 126 (2016) 364–373

I.H. Cho, M.H. Kim

fw = − ρω2ξ

0

∫−h [ϕ(1)( − a, z) − ϕ(3)(a, z)]dz

by user-deﬁned RPM inputs to the motor. The sway motion amplitude of the tank was controlled by positioning the end of a connecting bar at the proper place of a slot of rotating disk with diameter 20 cm. In the experiment, the tank was forced to oscillate horizontally with amplitude 0.3 cm for all oscillation periods. The period range was from 0.3 s to 1.4 s The dual vertical porous baﬄes were made of a punched steel plate with the thickness of 2 mm. The spacing of equally distributed circular holes was adjusted to give three diﬀerent porosities (P = 0.0567, 0.1275, 0.3265). Table 1 summarizes the principal speciﬁcations of the porous baﬄes used in our experiments. The submergence depths of the vertical porous baﬄes were d=5 cm,10 cm and the water depth was ﬁxed at 10 cm. The dual porous baﬄes were attached to the rectangular tank symmetrically with respect to the z -axis and placed at a1 = − 8.3 cm, a 2 = 8.3 cm , as tabulated in Table 2. For the 2-D rectangular tank without baﬄes, the lowest natural frequency of sloshing in the tank can be computed by the following formula (Ibrahim, 2005):

(21)

Decomposing fw into its real and imaginary parts gives,

fw = (ω2μ + iων )ξ

(22)

where μ is the added mass and ν the damping.

⎧ ⎫ In a similar way, the hydrodynamic forces (Fj = Re⎨f j e−iωt ⎬) on the ⎭ ⎩ baﬄes are as follows: f j = − ρω2ξ

0

∫−d [ϕ(j+1)(aj , z) − ϕ(j)(aj , z)]dz,

j = 1, 2

(23)

j

The wave elevations in a tank and dynamic pressures on tank wall can be explicitly expressed as follows:

η(x ) =

ω 2ξ ϕ(x, 0), p(z ) = ρω2ξϕ( ± a, z ) g

(24)

T= 3. Experiments

Table 1 Specification of porous baffles used in model tests. Porosity

Diameter of hole

Spacing of adjacent holes

Punched Plate

P=0.0567 P=0.1275 P=0.3265

3 mm

5 mm 8 mm 12 mm

, n = 1, 2, 3, ... (25)

where T is the natural period, and n=1,2,3,.. the sloshing modes. The natural periods of baﬄe-free sloshing for present experimental model are 1.07 s, 0.61 s, and 0.47 s The dynamic pressure acting on the tank wall was measured with a water-tight pressure gauge (P310-02S). The sampling frequency was 100 Hz. The pressure gauge was installed at the RHS wall and 2 cm apart from the tank bottom. The wave elevation at the wall was measured through the image processing of video recording with a iPhone 6 built-in camera(resolution: 1280×720, 240 frame/s). To improve optical detection, the water was dyed with red color. The photograph of the experimental set-up is shown in Fig. 2.

In order to validate the analytic solutions developed in the preceding section, we conducted a series of experiments with a small scaled model. The inner dimensions of the rectangular container are 50 cm×10 cm×50 cm and it is made of transparent acrylic resin with 1.2 cm thickness. The tank model was placed on the frictionless LM guide ﬁxed at the table. The tank was actuated within the period range

Type

2π nπg nπh ( 2a )tanh( 2a )

4. Numerical results and discussions 4.1. Comparison with experimental results In Exp. #1, two inner porous baﬄes with the same porosity are positioned at x = aj , (j = 1, 2) to the bottom of the tank. Fig. 3 and Fig. 4 show the change in the wave elevation at the left wall and dynamic pressure at the right wall as function of the motion period. The y-axis of Fig. 3 denotes the ampliﬁcation factor, which is determined as the wave elevation amplitude at the wall of the tank divided by the tank's motion amplitude (Ra = (ηmax − ηmin )/2ξ ). The yaxis in Fig. 4 is the pressure peak-to-trough diﬀerence ( pd = (pmax − pmin )) measured at the right-hand side of the wall 2 cm away from the tank's bottom. The solid lines refer to the MEEM analytic solutions and the marks(•) are the experimental results. When

Table 2 Experimental conditions in model tests. Exp.#1 Number of baﬄes Submergence depth of baﬄes Position of baﬄes Porosity Sway motion period Sway motion amplitudes

2 10 cm x = ± 8.3 cm 0.0567/0.1275/0.3265 0.3–1.4 s 0.3 cm

Exp.#2

5 cm

Fig. 2. Photograph of experimental set-up. (a) fully submerged (b) partially submerged.

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Ocean Engineering 126 (2016) 364–373

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16

pd

Ra

Exp.#1 MEEM solution

400

12

Exp.#1 MEEM soution

2

[N/m ]

300 8

200

4

100 0 0.4

0.6

0.8

1.0

1.2

1.4

0

T [sec]

0.4

0.6

(a) P=0.0567

0.8

1.0

1.2

1.4

T [sec]

(a) P=0.0567

10

Ra

pd 8

Exp.#1 MEEM solution

200

6

Exp.#1 MEEM solution

150 2

[N/m ]

4

100

2 50

0 0.4

0.6

0.8

1.0

1.2

1.4

T [sec]

0 0.4

(b) P=0.1275

0.6

0.8

1.0

1.2

1.4

1.2

1.4

T [sec]

10

(b) P=0.1275

Ra Exp.#1 MEEM solution

8

pd 200

6

Exp.#1 MEEM solution

150 2

[N/m ]

4

2

0 0.4

0.6

0.8

1.0

1.2

100

50

1.4

T [sec]

(c) P=0.3265

0 0.4

Fig. 3. Comparison of ampliﬁcation factor (Ra ) between the analytic solutions and experimental results for d /h = 1.0 .

0.6

0.8

1.0

T [sec]

(c) P=0.3265

the inner porous baﬄe is impermeable, the rectangular tank can be divided into three separate smaller tanks with the length 2c=16.7 cm. The sloshing natural periods of the smaller tanks in three consecutive modes (n = 1, 2, 3) are 0.47 s, 0.33 s, and 0.26 s, respectively. Fig. 3 shows the results of three porosities (P = 0.0567, 0.1275, 0.3265). When

Fig. 4. Comparison of dynamic pressure at tank wall between the analytic solutions and experimental results for d /h = 1.0 .

368

Ocean Engineering 126 (2016) 364–373

I.H. Cho, M.H. Kim

10

P = 0.0567, 0.1275, with relatively low porosity, the results exhibit resonance at the resonance period of T = 0.47 s. However, in the case of P = 0.3265 with signiﬁcant porosity, in addition to primary peak at 0.47 s as before, a minor secondary peak is also observed at 1.07 s, which is the lowest mode of the baﬄe-free tank. This is because although the inner porous baﬄes are fully extended to the tank's bottom, the ﬂuid can move through the inner porous baﬄes of large porosity and thus the lowest mode of the baﬄe-free tank starts to appear. It is also seen that the corresponding pressure curves have similar trends compared to the ampliﬁcation factors (Fig. 3), which is expected since pressure is directly related to the surface elevation in the linear potential theory. The integration of the local pressure over the wall surface will produce the total sloshing-induced force on the wall. The theoretically predicted curves correlate well with the experimental results within the range of experimental periods except at the resonant period (T = 0.47 s). The discrepancy between the two results at the resonant period can be attributed primarily to the nonlinear eﬀects which are ignored in the present linear potential theory and partially to viscosity (wall friction). Love and Tait (2010, 2013), with their onebaﬄe experiment, showed that the nonlinear free-surface eﬀect becomes increasingly more important as tank oscillation amplitude increases. Figs. 5 and 6 have the same conditions as Exp. #1 except that the inner porous baﬄes are only halfway submerged (Exp. #2). In this case, the ﬂuid moves freely under the lower part of the inner baﬄes especially for long-period motions. Therefore, regardless of the porosity of the inner porous baﬄes, resonances occur not only at 0.47 s but also near 1.07 s. The former is the lowest natural frequency of the narrow region or 3rd lowest mode of the baﬄe-free entire region. The latter is related to the lowest resonant motion of the entire baﬄe-free region. The actual resonance period near 1.07 s is shifted to slightly larger value due to diﬀraction eﬀects by inner baﬄes. The analytic result for P = 0.0567 (lowest porosity case) indicates that around the resonant period of 0.47 s, double peaks occur. The experimental data showing similar trend against prediction also indicate relevant viscous and nonlinear eﬀects especially near resonance. With regard to the resonance near 1.2 s, its peak is shifted to slightly longer period as baﬄe porosity decreases, as shown both in theory and measurement. This trend has also been reported in Faltinsen and Timokha work (2011). In the case of resonance near 1.2 s with low porosities (P = 0.0567, 0.1275), there is signiﬁcant discrepancy in the peak magnitudes between the analytical result and the experimental result, where nonlinear free-surface eﬀects play a crucial role. However, when the porosity is high (P = 0.3265), the discrepancy is greatly reduced. The small diﬀerences in higher-porosity inner baﬄes are mainly due to the viscous eﬀects while the liquid moves along the wall/baﬄes and around the edges of the baﬄes (e.g. Fig. 8), which are neglected in the present potential theory. Like the case of fully extended baﬄes, the corresponding pressure curves generally have similar trends as surface elevations (Fig. 5), as shown in Fig. 6. The above discussions can further be veriﬁed by introducing MPS (moving particle semi-implicit) computation. The MPS is a grid-less Lagrangian approach and solves Navier-Stokes equation with fullynonlinear free-surface conditions so that both viscous and nonlinear eﬀects can be included. The details of the MPS method can be found in the second author's prior publications (e.g. Kim et al., 2011; 2014). In the CFD modeling, since many holes on baﬄes are diﬃcult to be modeled by the MPS method, only the zero-porous-half-submerged-baﬄe case is considered and its results are compared in Fig. 7 to the counter part of linear potential theory at 0.47 s and 1.07 s, which are baﬄe-free resonance frequencies. Its potential-theory result resembles with that of the lowest-porosity case (Fig. 5a), as expected. The total ﬂuid particles used are 7858. For rigid surfaces, 1044 surface particles and 2336 dummy particles were used. We can see that the peak at 0.47 s is signiﬁcantly limited by viscous and nonlinear free-surface eﬀects. To

Ra

Exp.#2 MEEM solution

8

6

4

2

0 0.4

0.6

0.8

1.0

1.2

1.4

1.2

1.4

1.2

1.4

T [sec]

(a) P=0.0567 10

Ra

Exp.#2 MEEM solution

8

6

4

2

0 0.4

0.6

0.8

1.0

T [sec]

(b) P=0.1275 10

Ra 8

Exp.#2 MEEM solution

6

4

2

0 0.4

0.6

0.8

1.0

T [sec]

(c) P=0.3265 Fig. 5. Comparison of ampliﬁcation factor (Ra ) between the analytic solutions and experimental results for d /h = 0.5 .

observe this case in more details, the actual ﬂow velocity vectors (at 0.47 s) obtained from the MPS method are plotted in Fig. 8. The signiﬁcant lowest sloshing mode is seen in each narrow region. We can 369

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16

pd 200

Ra

2

[N/m ]

12

Exp.#2 MEEM solution

150

MEEM solution MPS solution baffle-free tank

8 100

4 50

0

0 0.4

0.6

0.8

1.0

1.2

0.4

1.4

0.6

0.8

1.0

1.2

1.4

T [sec]

T [sec]

Fig. 7. The comparison between linear potential theory and MPS calculation for the halfsubmerged zero-porosity baﬄe.

(a) P=0.0567

pd 200

Exp.#2 MEEM solution

2

[N/m ]

150

100

50

Fig. 8. Velocity vector snapshot for the case of Fig. 7 at 0.47 s by MPS method.

also observe the rotational ﬂows around the bottoms of the baﬄes which increase the corresponding viscous eﬀects. From the above discussions, the linear potential theory with the present porosity model can capture the essential physics of the sloshing with baﬄes so that it can be repeatedly used for its early-stage design.

0 0.4

0.6

0.8

1.0

1.2

1.4

T [sec]

(b) P=0.1275

pd

4.2. Design of optimal baﬄe

250

Extensive calculations have been conducted to ﬁnd the best design of dual vertical porous baﬄes with varying design parameters, such as installation position, porosity, and the submerged depth of the baﬄe. Fig. 9 shows the ampliﬁcation factors at the tank's wall when the location of the inner porous baﬄe is varied. The x-axis is the nondimensional frequency of the tank motion ω2h / g . The submerged depth of the baﬄe in this case is equal to the tank's water depth and the baﬄe porosity is ﬁxed at 0.1. The increase in the c / a value implies that the porous baﬄe is located closer to the wall. Within the considered frequency range, the general tendency is that the ampliﬁcation factor decreases as c / a increases. However, if baﬄes are too close to the wall, it can be worse. Among the four values of c / a , the best selection for the smallest ampliﬁcation ratio at the wall is when c / a = 0.8 with the given porosity. Fig. 10 exhibits the non-dimensional added mass (μ¯ = μ /2agh ) and damping coeﬃcient (ν¯ = ν /2ωagh ) when c / a = 0.8 under the same condition as Fig. 9. When the resonant frequency ω2h / g = 0.35, the non-dimensional added mass is negative, and the damping coeﬃcient reaches its peak value. This unique phenomenon usually occurs in cases when the moving structure excites local hydrodynamic resonance in its vicinity (McIver and Evans, 1984). Related examples are oﬀshore structures with moon-pools, catamarans, ships moored at quay walls etc.

Exp.#2 MEEM solution

200

2

[N/m ]

150

100

50

0 0.4

0.6

0.8

1.0

1.2

1.4

T [sec]

(c) P=0.3265 Fig. 6. Comparison of dynamic pressure at tank wall between the analytic solutions and experimental results for d /h = 0.5.

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Ra

f ρ gd ξ

c/a=0.4 c/a=0.6 c/a=0.8 c/a=0.9

14 12

16

10

14

8

12

6

10

4

8

2

c/a=0.4 c/a=0.6 c/a=0.8 c/a=0.9

6

0 0

1

2

3

4

4

2

ω h/g

2

Fig. 9. Ampliﬁcation factor(Ra ) at tank wall as a function of arrangement of dual porous baﬄes for a /h = 2.5, d /h = 1.0, P = 0.1, ξ /h = 0.025.

0

0

1

2 2

3

4

ω h/g Fig. 11. Horizontal forces on each porous baﬄe as a function of arrangement of dual porous baﬄes for a /h = 2.5, d /h = 1.0, P = 0.1, ξ /h = 0.025.

0.5 0.4

added mass damping coeff.

0.3

20

Ra

0.2

d/h=0.25 d/h=0.5 d/h=1.0

15

0.1 0.0

10

-0.1 -0.2 5

-0.3 0

1

2

3

4

ω h/g 2

0

0

Fig. 10. Non-dimensional added mass ( μ¯ ) and damping coeﬃcient (ν¯ ) by sway motion of a rectangular tank with dual porous baﬄes for a /h = 2.5, d /h = 1.0, P = 0.1, c /a = 0.8, ξ /h = 0.025.

1

2

3

4

2

ω h/g Fig. 12. Ampliﬁcation factor (Ra ) at tank wall by sway motion of a rectangular tank as a function of submergence depth of dual porous baﬄes for a /h = 2.5, P = 0.1, c /a = 0.8, ξ /h = 0.025.

Fig. 11 shows the corresponding horizontal forces on the two inner porous baﬄes according to the location of the baﬄes as in Fig. 9. Since the inner porous baﬄes are installed symmetrically with respect to the z-axis, both baﬄes have the same horizontal forces ( f = f j , j = 1, 2 ). Similar to the wall ampliﬁcation factor, less horizontal force is applied as baﬄes are closer to the wall. When c / a = 0.4 , horizontal forces become very large at the resonance frequency of ω2h / g = 2.0 . When c / a = 0.6 , horizontal forces are large at ω2h / g = 3.0 . When the baﬄe is located closest to the wall (c / a = 0.8, 0.9), the number of resonance frequencies increases but peak amplitudes become smaller compared to the cases of c / a = 0.4, 0.6 . Fig. 12 shows the change of wall ampliﬁcation factors as the submerged depth of the baﬄe is varied while its location is ﬁxed at c / a = 0.8. When submerged depth is the shallowest at d / h = 0.25, among the natural sloshing frequencies ω2h / g = 0.35, 1.1, 1.8 of the baﬄe-free tank, high ampliﬁcations can be observed at the ﬁrst and third modes. The second mode has its node at the wall, so no peak appears there. As baﬄe's submergence depth increases, the ﬂuid region is more separated by the baﬄe, so the peak near ω2h / g = 1.8 disappears

ﬁrst. Further increase of the baﬄe's submergence depth eventually shuts down any crossﬂow among each narrow ﬂuid regions and diminishes the lowest-mode peak at ω2h / g = 0.35. Interestingly, the half-submergence case is as good as the full-submergence case except for the narrow resonance peak at ω2h / g = 0.35, which will anyway be limited by the nonlinear and viscous eﬀects. Lastly, Fig. 13 shows the trend of wall ampliﬁcation factors with changing baﬄe's porosity. This case is for the dual baﬄes fully extended to the tank bottom with c / a = 0.8. When the baﬄe porosity increases, higher resonance peaks occur since ﬂuid can move more freely through the baﬄe and the baﬄe's restriction eﬀect is minimized. So, when only the wall ampliﬁcation factor is considered, the lowest porous baﬄe seems the most eﬀective. However, the horizontal forces on baﬄes, as illustrated in Fig. 14, increase as porosity decreases, as can be intuitively expected i.e. there is a trade-oﬀ between wall ampliﬁcation factors and forces on baﬄes. Therefore, both the ampli371

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Ra

both nonlinear free-surface and viscous eﬀects. In this regard, the linear potential theory can be used for the initial design and optimization of such baﬄes. The presence of vertical baﬄes shifts the baﬄe-free sloshing natural periods, especially in the lowest mode. As baﬄe porosity (P ) increases and its submergence depth (d / h ) decreases, the ﬂuid can move more freely across the baﬄes. Correspondingly, high ampliﬁcation factor was observed at resonance frequencies and the natural sloshing frequencies got closer to those of the baﬄe-free tank. It is also seen that the pressure and sloshing force on tank wall are closely related to the change of wall ampliﬁcation factor. Also, as the baﬄes get closer to the wall, the ampliﬁcation factor and horizontal force on baﬄes become smaller although negative eﬀects occur if they are too close to the wall. When only the wall ampliﬁcation factor is considered, the lowest porous baﬄe seems the most eﬀective. However, the horizontal forces on baﬄes increase as porosity decreases i.e. there is a trade-oﬀ between wall ampliﬁcation factors and forces on baﬄes. Therefore, both the ampliﬁcation factor and the horizontal force need to be simultaneously examined to ﬁnd the optimal porosity for the given case.

P=0.05 P=0.1 P=0.2 P=0.3

8

6

4

2

0 0

1

2

3

4

2

ω h/g Fig. 13. Ampliﬁcation factor (Ra ) at tank wall by sway motion of a rectangular tank as a function of porosity of dual porous baﬄes for a /h = 2.5, d /h = 1.0, c /a = 0.8, ξ /h = 0.025.

Acknowledgment This research was supported by the 2016 scientiﬁc promotion program funded by Jeju National University , South Korea.

f ρ gd ξ

References

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P=0.05 P=0.1 P=0.2 P=0.3

5

4

3

2

1

0 0

1

2

3

4

2

ω h/g Fig. 14. Horizontal forces on porous baﬄes as a function of porosity of dual porous baﬄes for a /h = 2.5, d /h = 1.0, c /a = 0.8, ξ /h = 0.025.

ﬁcation factor and the horizontal force need to be simultaneously examined to ﬁnd the optimal porosity for the given case. 5. Concluding remarks In this study, the analytic solutions for the sloshing inside a rectangular tank with dual porous baﬄes of arbitrary submergence depth were derived by means of matched eigenfunction expansion method in the context of linear potential theory. The analytical results are then compared with the experiments conducted by authors. The inner porous baﬄes are intended for minimizing liquid sloshing responses with reduced forces on them and tank walls. The linear potential theory agrees very reasonably with the experimental results and MPS computation when forced tank oscillations are not large. It also captures the resonance frequencies correctly while their peaks are exaggerated. The signiﬁcant discrepancy in the peak magnitudes between the analytical and experimental results can be attributed to 372

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Liu, Y., Li, Y.C., Teng, B., 2008. Wave motion over a submerged breakwater with an upper horizontal porous plate and a lower horizontal solid plate. Ocean Eng. 35, 1588–1596. Love, J.S., Tait, M.J., 2010. Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique. J. Fluids Struct. 26 (7–8), 1058–1077. Love, J.S., Tait, M.J., 2013. Parametric depth ratio study on tuned liquid dampers: ﬂuid modelling and experimental work. Comput. Fluids 79, 13–26. McIver, P., Evans, D.V., 1984. The occurrence of negative added mass in free-surface problems involving submerged oscillating bodies. J. Eng. Math. 18, 7–22. Mei, C.C., 1989. The applied dynamics of ocean surface wavesAdvanced Series on Ocean Engineering vol. 1. World Scientiﬁc, Singapore. Mei, C.C., Liu, P.L.F., Ippen, A.T., 1974. Quadratic head loss and scattering of long waves. J. Waterw. Harb. Coast. Eng. Div. 99, 209–229. Molin, B., Remy, F., 2013. Experimental and numerical study of the sloshing motion in a

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