Experimental and numerical study of the sloshing motion in a rectangular tank with a perforated screen

Experimental and numerical study of the sloshing motion in a rectangular tank with a perforated screen

Journal of Fluids and Structures 43 (2013) 463–480 Contents lists available at ScienceDirect Journal of Fluids and Structures journal homepage: www...

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Journal of Fluids and Structures 43 (2013) 463–480

Contents lists available at ScienceDirect

Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs

Experimental and numerical study of the sloshing motion in a rectangular tank with a perforated screen Bernard Molin n, Fabien Remy Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13451 Marseille cedex 20, France

a r t i c l e i n f o

abstract

Article history: Received 30 January 2013 Accepted 5 October 2013 Available online 28 October 2013

Rectangular tanks partially filled with water and fitted with vertical perforated screens have been proposed as Tuned Liquid Dampers to mitigate the vibratory response of land buildings, under wind or earthquake excitation. Similar devices are used as anti-rolling tanks aboard ships. Experiments are performed on a rectangular tank with one screen at mid length. The tank is subjected to forced horizontal and rolling motions, harmonic and irregular. The open-area ratio of the screen is kept constant while the motion amplitudes and frequencies are varied. The frequency range covers the first three natural sloshing modes of the clean tank (without screen). Force measurements are converted into matrices of added mass/inertia and damping coefficients. A simple numerical model is proposed, based on linearized potential flow theory and quadratic discharge equation at the screen, following earlier works by the first author. Good agreement is reported between experimental and numerical hydrodynamic coefficients. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Tuned liquid damper (TLD) Sloshing Perforated screen Potential flow theory Hydrodynamic coefficients

1. Introduction In the recent past much work has been published on the so-called TLDs, or Tuned Liquid Dampers. We refer readers to the recent papers by Tait (2008), Faltinsen et al. (2011), or Crowley and Porter (2012a, 2012b) where extensive reviews can be found. These TLDs, which consist of rectangular tanks, fitted with one or more vertical perforated screens, and filled with water to an appropriate level, serve the purpose of mitigating the vibratory response of slender buildings under wind or earthquake excitation. This vibratory response, at the roof level where TLDs are usually located, mostly takes place in the horizontal direction; hence most of the TLD literature deals with forced translatory motion. In ship design, there are parent devices, known as anti-rolling tanks, which are meant to reduce the roll response under wave excitation. Vertical screens have also been considered to reduce the roll and sway induced sloshing motion in LNG or oil tanks. In land building applications TLDs are tuned in the sense that the natural frequency of their first sloshing mode is made to coincide with the natural frequency of the vibratory mode which they are meant to damp, and which is invariable after construction. On the other hand roll natural frequencies of marine structures are susceptible to change due to varying loading conditions. The question therefore arises of the efficiency of the anti-rolling tank when the roll and tank natural frequencies are separated. Various numerical models have been proposed to assess the performance of TLDs, most of these models being based on potential flow theory. Love and Tait (2010) apply the multi-modal approach of Faltinsen and Timokha (2001) with the

n

Corresponding author. Tel.: þ33 49 105 4641. E-mail addresses: [email protected] (B. Molin), [email protected] (F. Remy).

0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.10.001

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nonlinear free surface equations taken to order one (linearized), three or five in the wave steepness, and the effects of the screens accounted for via a quadratic damping term introduced in the modal equations. Faltinsen et al. (2011) comment that this method applies under the condition that the screen solidity ratio Sn be small, i.e. that the screen be relatively transparent to the flow. As its solidity increases a jump in the free surface elevation arises at the screen, meaning that the truncation order of the modal expansion must be taken higher and higher. [It must be commented here that, as shown in Molin (2011), and as will be made clear further on, the relevant parameter is not the solidity ratio Sn, but a combination of solidity ratio, motion amplitude and tank length under the form A Sn =½ð1  Sn Þ2 b with A the tank motion amplitude and b its half-length.] Faltinsen et al. (2011) propose a variant of their multimodal theory that accounts for slatted screens of high solidity ratios, with the free surface equations linearized and quadratic pressure losses at the mid-tank screen. They report extensive experiments with a rectangular tank 1 m long, at a filling height of 40 cm, undergoing harmonic sway motion over a frequency range encompassing the first three natural frequencies of the sloshing modes. Eight slatted screens of varying solidity ratios, from 0.4725 through 0.95125, are successively tested. Only the wave elevations, by the end walls, are measured. Good agreements between measured and calculated free surface elevations are reported at the smallest sway amplitude of 1 mm. At 10 mm sway amplitude discrepancies start appearing due to secondary resonances. The main purpose of this paper is to complement Faltinsen et al.'s results by performing experiments with a rectangular tank of identical depth over length ratio (length 80 cm, water height 32 cm), where the main focus is the measurement of the hydrodynamic loads. In contrast to their experiments, the open-area ratio of the perforated screen is kept constant (18%) and the motion amplitude is varied. The tank is successively subjected to forced motions in sway and roll and the measured hydrodynamic loads are converted as added mass/inertia Caij and damping Cbij coefficients, with i, j being 2 or 4 (2 for sway and 4 for roll). A simple numerical model is proposed, based on previous work by the first author and his colleagues (Molin and Fourest, 1992; Kimmoun et al., 2001; Molin, 2011). Good agreement is found between numerical and experimental hydrodynamic coefficients in all cases where no breaking of the free surface was observed. The contents of the paper are as follows: the experimental set-up is first described. Then the numerical model is presented and its main differences with the models of Faltinsen et al. (2011) and Crowley and Porter (2012a, 2012b) are outlined. Forced sway motion is first considered, with the tank being oscillated at amplitude over length ratios ranging from 0.0025 through 0.0150, and frequencies encompassing the first three natural modes. Some tests with irregular imposed motion are also performed. Then force harmonic roll motion is considered, with roll induced sway motion at free surface level in the same range as in the pure sway cases. It is found that the experimental hydrodynamic coefficients agree well with the numerical ones, except at the larger motion amplitudes, around the natural frequency of the second mode, where free surface breaking occurs. 2. Experimental set-up The experiments were performed with the Mistral Hexapode test bench from Symetrie,1 shown in Fig. 1. This bench is fitted with force sensors which deliver, after some processing, the complete force tensor due to the liquid motion inside the tank. The tank has an inner length of 80 cm and an inner width of 50 cm, for a height of about 60 cm. In the experiments reported here it was filled with water to a height of 32 cm, that is the same waterdepth over tank length ratio as in the experiments reported in Faltinsen et al. (2011). The tank was subdivided in 4 compartments, 80 cm long and about 12 cm wide, in order to minimize the effects of transverse sloshing. This set-up also ensured that the perforated plates remained rigid. The perforated screen, set at mid-tank, consisted of 4 steel plates (one per compartment), 2 mm thick, with circular openings of diameter 4 mm (Fig. 2). The open-area ratio τ ¼ 1  Sn was 18%. It must be noted here that the screens modeled by Faltinsen et al. (2011) are different since they consist in a succession of horizontal slots and slats. However in Molin and Legras (1990), it is demonstrated through experiments on “stabilizers” that quasi-identical hydrodynamic coefficients are obtained, whether the openings consist in circular holes or slots, provided the open-area ratios are the same. In some tests the free surface elevation at one of the end walls was measured through image processing of video recordings with a camera attached to the Hexapode. To improve optical detection the water was dyed with fluorescein. The sampling frequency was 25 Hz while for the load measurement it was 100 Hz. 3. Theoretical/numerical model The theoretical model is based on potential flow theory, with the free surface equations linearized. Similar approaches are used by Crowley and Porter (2012a, 2012b) and Faltinsen et al. (2011). The main difference is that we represent the screen as a porous boundary, while Crowley and Porter and Faltinsen et al. account for the exact geometry of the screen (in their case a succession of slots and slats, all over the height). 1

http://www.symetrie.fr/en/

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

465

Fig. 1. The tank on ECM Mistral Hexapode.

Fig. 2. Perforated screens.

We take h as the liquid depth and 2b as the length of the tank in the y direction (see Fig. 3). The coordinate system Oyz is centered at mid-tank, with z¼0 the undisturbed free surface and y¼0, h r z r0, the screen. However the roll axis of rotation is taken at the tank bottom level (in y¼0, z ¼ h) and the roll moment is referred to this position. 3.1. Forced harmonic motion in sway Let vðtÞ ¼ Aω cos ωt be the imposed sway velocity. We look for a steady-state solution, with the velocity potential written as Φðy; z; tÞ ¼ Rfφðy; zÞ e  iωt g:

ð1Þ

The velocity potential φ satisfies the Laplace equation in b r yr b h r z r0, the linearized free surface equation gφz  ω2 φ ¼ 0 at z¼0, and no flow conditions at the solid walls: φy ð 7b; zÞ ¼ Aω;

φz ðy; hÞ ¼ 0:

ð2Þ

At the porous wall the pressure drop P  P þ is written as P  P þ ¼ ρ

1τ V r jV r j; 2μτ2

ð3Þ

with τ the open-area ratio, μ a discharge coefficient (μ C 0:5 for perforated screens), and Vr the relative flow velocity in the normal direction to the porous screen (e.g. see Molin, 2011).

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Fig. 3. Geometry.

The velocity potential is looked for under the form 1

φðy; zÞ ¼ Aωy þ ∑ Am cos λm ðy bÞ m¼1

7B0 cos k0 ðy 7bÞ

cosh λm ðz þhÞ cosh λm h

1 cosh k0 ðz þ hÞ cosh kn ðy 7 bÞ 7 ∑ Bn cos kn ðz þhÞ; cosh k0 h cosh kn b n¼1

ð4Þ

where λm ¼ ð2m  1Þπ=ð2bÞ, Am ¼ 2Aω3 =ðλ2m ðω2m  ω2 ÞbÞ, ω2m ¼ gλm tanh λm h and ω2 ¼ g k0 tanh k0 h ¼  g kn tan kn h. Where 7 appears, it means that the þ sign is to be used in the left-hand side compartment of the tank and the  sign in the right-hand side compartment. In this way continuity of the horizontal velocity at the screen is ensured. When there is no screen B0  Bn  0. Note that the Am coefficients are real whereas the Bn coefficients are complex. With φ given as (4) the Laplace equation and all boundary conditions are satisfied except for the discharge equation at the porous screen. With the linearized hydrodynamic pressure being P ¼ ρ∂Φ=∂t ¼ Rfiρωφexpð iωtÞg and Lorentz linearization being applied, the discharge equation takes the form     4 1τ iω φ   φ þ ¼ J φy  Aω J φy Aω ; 3π μτ2

ð5Þ

where J J denotes the modulus of the complex number. This is satisfied through iterations by writing Eq. (5) under the form   4 1 τ ðj  3=2Þ ðjÞ J φy  AωJ φðjÞ iωðφðjÞ  φþ Þ ¼ y  Aω ; 3π μτ2

ð6Þ

which leads to   cosh k ðz þhÞ ðj  3=2Þ 0 BðjÞ cos k0 b þ k0 sin k0 bf ðzÞ 0 cosh k0 h þ∑BðjÞ n ð1  kn tanh kn bf

ðj  3=2Þ

n

ðzÞÞ cos kn ðz þhÞ ¼ f

ðj  3=2Þ

ðzÞgðzÞ;

ð7Þ

with g ðzÞ ¼ ∑λm ð  1Þðm þ 1Þ Am m

ðjÞ

f ðzÞ ¼ 

cosh λm ðz þ hÞ ; cosh λm h

   2i 1  τ  g ðzÞ k0 BðjÞ sin k0 b cosh k0 ðz þ hÞ þ ∑kn BðjÞ tanh kn b cos kn ðz þhÞ: n  0 3πω μτ2  cosh k0 h n

ð8Þ

ð9Þ

In Eqs. (6) and (7) ðj 3=2Þ means that the averaged values of Bn between the previous two iterations ðj  2Þ and ðj  1Þ are used to compute f(z). In this way relaxation is introduced in the iterative scheme and convergence is faster. The Am and Bn series are truncated at orders M and N respectively and the following linear system is built up Z 0 Z 0 2 ðj  3=2Þ cosh k0 ðz þ hÞ 1 kn tanh kn bf ðzÞ ðjÞ dz þ ∑ BðjÞ B cos kn ðz þ hÞcosh k0 ðz þ hÞ dz n 0 ðj  3=2Þ cosh k h n 0 h  h cos k0 b þk0 sin k0 bf ðzÞ Z 0 ðj  3=2Þ f ðzÞgðzÞ ¼ ð10Þ cosh k0 ðz þ hÞ dz; ðj  3=2Þ  h cos k0 b þk0 sin k0 bf ðzÞ BðjÞ 0

Z

0

ðj  3=2Þ

ðzÞ cosh k0 ðz þ hÞ cos km ðz þ hÞ dz cosh k0 h ðzÞ 1  km tanh km bf Z 0 Z 0 ðj  3=2Þ 1  kn tanh kn b f ðzÞ þBðjÞ cos 2 km ðz þhÞ dz þ ∑ BðjÞ cos km ðz þ hÞ cos kn ðz þ hÞ dz m n ðj  3=2Þ nam h  h 1  km tanh km b f ðzÞ

h

cos k0 bþ k0 sin k0 bf

ðj  3=2Þ

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

Z ¼

0 h

f

ðj  3=2Þ

ðzÞgðzÞ

1  km tanh km bf

ðj  3=2Þ

ðzÞ

cos km ðz þ hÞ dz;

467

ð11Þ

and solved with a Gauss routine. Convergence is reached within 10–20 iterations with the Bn coefficients initially taken equal to zero. With the Am and Bn series truncated after 10 terms, 4 digits accuracy is obtained in the cases reported here. The velocity potential at the wall, at the free surface, is then obtained as M

N

m¼1

n¼1

φð b; 0Þ ¼  Aωb ∑ Am þ B0 þ ∑ Bn

cos kn h ; cosh kn b

ð12Þ

and the free surface elevation Response Amplitude Operator (RAO) is ωJ φð  b; 0Þ J =ðAgÞ (the RAO being defined as the amplitude of the free surface motion divided by the sway motion amplitude). The sway hydrodynamic load is obtained by integrating the dynamic pressure iωρφ on the solid walls plus perforated screen. It consists in a component that opposes the sway acceleration and a component that opposes the sway velocity: F y ¼ Rf2iρAω2 bhðC a22 þ i C b22 Þe  iωt g;

ð13Þ

with tanh λm h tanh k0 h þ B0 ð cos k0 b  1Þ λm k0   N 1 sin kn h þ ∑ Bn 1  ; cosh kn b kn n¼1 M

Aωbh½C a22 þi C b22  ¼ Aωbh þ ∑ Am m¼1

ð14Þ

Ca22 being the added mass coefficient and Cb22 the damping coefficient. Likewise the hydrodynamic moment with respect to the mid-bottom point can be obtained analytically from the Am and 2 3 Bn coefficients, made non-dimensional with iρAω2 ðbh þ2b =3Þ and expressed via coefficients Ca42 and Cb42. It should be noted that, under our proposed model, for a given tank and given oscillation frequency, the hydrodynamic coefficients depend on the open-area ratio τ, discharge coefficient μ, motion amplitude A and tank half-length b only through the coefficient Að1 τÞ=ðμτ2 bÞ: it is equivalent to vary the motion amplitude or to vary the porosity as long as this coefficient is kept constant. This statement holds under the condition that the linear free surface equations remain valid.

3.1.1. Comments with regards to other numerical models Crowley and Porter (2012a, 2012b) use a method somewhat similar to ours, with the discharge equation (our Eq. (5)) at the wall under the form   ð15Þ φ   φ þ ¼ kγ0 φy  Aω ; where γ ¼ Cþ i KL , C being a blockage coefficient as predicted from potential flow theory (accounting for the exact geometry of the screen consisting in a succession of slots and slats) and KL a linearized drag coefficient which they determine through iterations. As shown in Molin (2011), the blockage coefficient from potential flow theory is nil in the limit when the number of openings increases to infinity (no matter the value of the open-area ratio). The main difference between Crowley and Porter (2012a, 2012b) and this paper is that their drag coefficient is averaged over the depth, whereas in our formulation it is z-dependent. It is expected that it could make appreciable differences at high oscillation frequencies when the depth variation of the flow kinematics is important. Faltinsen et al. (2011) has much similarity with Crowley and Porter (2012a, 2012b) except that they use a time domain approach to solve the problem whereas Crowley and Porter, like us, use a frequency domain approach. From our experience the time domain approach ineluctably fails as the porosity τ goes to zero (the time derivative of the potential differential at the screen becomes infinite). The frequency domain approach that we use permits to cover all values of the parameter Að1  τÞ=ðμτ2 bÞ, from zero (no screen) to infinity (solid screen). Some authors (e.g. Warnitchai and Pinkaew, 1998, or Tait, 2008) assume that the first sloshing mode is dominant, so their approaches are limited to small values of the parameter Að1  τÞ=ðμτ2 bÞ and to a narrow range of frequencies around the first sloshing mode. Our model can tackle all values of Að1 τÞ=ðμτ2 bÞ and of the imposed frequency.

3.2. Forced harmonic roll motion We take the roll axis at mid bottom in y¼0, z¼  h, and  A the induced horizontal motion at the free surface level due to a positive (anticlockwise) rotation. When there is no screen the velocity potential can be obtained as

1 Aω 2 sinh λm z cosh λm ðz þ hÞ  yðz þhÞ þ ∑ αm cos λm ðy bÞ φNS ðy; zÞ ¼ þβm ; ð16Þ h λm cosh λm h cosh λm h m¼1

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where αm ¼ 

8b ð2m  1Þ2 π 2

;

ð17Þ

10

no wall 1 mm 2 mm 4 mm 8 mm 16 mm solid wall

9 8 7 6 5 4 3 2 1 0

3

4

5

6

7

8

9

10

11

12

13

frequency (rad/s) Fig. 4. Harmonic sway motion. RAO of the free surface elevation at the wall. Results from computations.

A=2mm

10 9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

3

4

5

6

7

8

A=6mm

10

9

10

11

12

13

0 3

4

5

6

7

ω rads−1 A=10mm

10

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1 3

4

5

6

7

8

ω rads−1

9

10

11

12

13

9

10

11

12

13

A=14mm

10

9

0

8

ω rads−1

9

10

11

12

13

0 3

4

5

6

7

8

ω rads−1

Fig. 5. Harmonic sway motion. RAO of the free surface elevation at the wall. Motion amplitudes 2 mm (top left), 6 mm, 10 mm and 14 mm (bottom right). Comparison between calculated (full lines) and measured values (lines with symbols).

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

βm ¼

469



1 ω2 ω2m

2g  g þω2 h : cosh λm h

ð18Þ

With the screen, alike in the previous case, the velocity potential is expressed as φðy; zÞ ¼ φNS ðy; zÞ 7 B0 cos k0 ðy7 bÞ

1 cosh k0 ðz þhÞ cosh kn ðy 7bÞ 7 ∑ Bn cos kn ðz þhÞ: cosh k0 h cosh kn b n¼1

ð19Þ

The discharge equation at the screen takes the form       4 1 τ  φy þ Aω z þ h φy þ Aω z þh : iω φ   φ þ ¼   3π μτ2 h h

ð20Þ

The resolution method is then identical with the sway case. In the end the added mass (inertia) and damping coefficients Ca24, Ca44, Cb24, Cb44 are obtained analytically from the Am and Bn coefficients. Note that the coefficients Ca44 and Cb44 shown 3 3 in the figures are made non-dimensional with reference to the solid inertia 2=3ρðbh þ b hÞ.

3.3. Irregular sway motion In the sway case, some tests were carried out with an irregular input motion, defined from an energy spectrum. 2.5

no wall 1 mm 2 mm 4 mm 8 mm 16 mm 32 mm solid wall

2

1.5

1

0.5

0

-0.5

-1

3

4

5

6

7

8

9

10

11

12

13

frequency (rad/s) Fig. 6. Harmonic sway motion. Calculated added mass coefficients Ca22 for sway amplitudes from 1 mm to 32 mm.

2.5

1 mm 2 mm 4 mm 8 mm 16 mm 32 mm

2

1.5

1

0.5

0

3

4

5

6

7

8

9

10

11

12

13

frequency (rad/s) Fig. 7. Harmonic sway motion. Calculated damping coefficients Cb22 for sway amplitudes from 1 mm to 32 mm.

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B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

The so-called “stochastic linearization” is then applied to the quadratic term in the discharge equation   Φy ð0; z; t Þ  V ðt Þ jΦy ð0; z; t Þ  V ðt Þj C

rffiffiffi   8 svr ðzÞ Φy ð0; z; t Þ  V ðt Þ ; π

A=2mm

3

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

3

4

5

6

7

8

A=4mm

3

2.5

−1

ð21Þ

9

10

11

12

13

−1

3

4

5

6

7

A=6mm

3

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5 −1 3

4

5

6

7

8

9

10

11

12

13

3

4

5

6

7

ω rads

ω rads

A=12mm

3 2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1 4

5

6

7

8

A=10mm

2.5

3

8

ω rads−1

10

11

12

13

9

10

11

12

13

9

10

11

12

13

−1

−1

3

9

A=8mm

3

2.5

−1

8

ω rads−1

ω rads−1

9

10

11

12

13

3

4

5

6

7

8

ω rads−1

Fig. 8. Harmonic sway motion. Measured and calculated sway added mass coefficient Ca22 vs. frequency ω. From top to bottom, and left to right: motion amplitudes of 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm.

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

471

with V(t) the imposed sway velocity and svr ðzÞ the standard deviation of the relative velocity at the screen, obtained from Z 1 s2vr ðzÞ ¼ jRAOvr ðz; ωÞj2 SðωÞ dω; ð22Þ 0

RAOvr ðz; ωÞ being the transfer function of the relative velocity at z and SðωÞ the input spectrum of the sway motion. The problem is then solved in the same fashion as in the harmonic case, frequency per frequency, while iterating over the value of the standard deviation svr ðzÞ. A=2mm

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

3

4

5

6

7

8

A=4mm

3

9

10

11

12

13

0

3

4

5

6

7

A=6mm

3

2.5

2

2

1.5

1.5

1

1

0.5

0.5

3

4

5

6

7

8

9

10

11

12

13

0

3

4

5

6

7

ω rads−1

2.5

2

2

1.5

1.5

1

1

0.5

0.5

3

4

5

6

7

8

ω rads−1

11

12

13

8

9

10

11

12

13

9

10

11

12

13

A=12mm

3

2.5

0

10

ω rads−1

A=10mm

3

9

A=8mm

3

2.5

0

8

ω rads−1

ω rads−1

9

10

11

12

13

0

3

4

5

6

7

8

ω rads−1

Fig. 9. Harmonic sway motion. Measured and calculated sway damping coefficient Cb22 vs. frequency ω. From top to bottom, and left to right: motion amplitudes of 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm.

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B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

4. Comparisons between experimental and numerical results 4.1. Forced sway. Harmonic motion 4.1.1. Free surface elevation at the wall Fig. 4 shows calculated Response Amplitude Operators (RAOs) of the free surface elevation at the end wall, vs. the oscillation frequency ω, for sway amplitudes ranging from 1 mm to 16 mm. [In all calculations shown in this paper, based on our experience with perforated plates, the discharge coefficient μ was taken equal to 0.5.] The cases where the screen is absent (porosity τ equal to 1) or solid ðτ  0Þ are also shown. Three peaks are visible, at the frequencies of the first three natural sloshing modes, ω1 ¼ 5:72 rad=s, ω2 ¼ 8:72 rad=s, and ω3 ¼ 10:74 rad=s. At small motion amplitude, alike in the no screen case, the RAO peaks at ω1 and ω3. As the motion amplitude increases these two peaks subside while the RAO gradually peaks at ω2. This is similar to what is reported in the experiments (and computations) of Faltinsen et al. (2011); in their case it is the screen porosity which is gradually decreased, while the amplitude is kept constant. As argued in the previous section, under our assumptions of linear free surface equations and quadratic pressure loss at the screen, the only relevant parameter is a combination of porosity and amplitude under the form Að1 τÞ=ðμτ2 Þ and it is equivalent to increase the amplitude or to decrease the porosity. It is striking that all RAO curves seem to intersect at the same two frequencies of 7.05 rad/s and 10.3 rad/s. Close inspection reveals that they do not exactly cross at the same frequencies. Fig. 5 shows a comparison between the measured and calculated RAOs, for motion amplitudes of 2 mm, 6 mm, 10 mm and 14 mm. The measured values are obtained from Fourier analysis of the time series, as their first harmonic, after a steady state has been reached. At 2 mm amplitude the measured values are somewhat lower than the calculated ones; we believe that this discrepancy is due to limited accuracy of the optical tracking system that we used: in this series of tests the motion amplitude of the free surface is only in the range of 1–3 mm. At 6 mm the agreement between computed and experimental RAOs is quite good. At 10 mm and 14 mm the numerical peak at the second sloshing frequency is much higher than the experimental one. This is associated with breaking of the free surface: in the 10 mm case the numerical RAO peaks at 7; this would mean a steepness 2ζ=λ2 (ζ being the free surface motion amplitude, and λ2  2b the wavelength associated with mode 2) equal to 0.175, much higher than the limiting steepness of standing waves as observed experimentally (e.g. see Taylor, 1953).

4.1.2. Hydrodynamic coefficients Figs. 6 and 7 show the calculated added mass and damping coefficients Ca22 and Cb22 for sway amplitudes ranging from 1 to 32 mm. Added mass coefficients are also shown in the cases when the screen is absent (τ ¼ 1 or, equivalently, A C0) or completely solid (τ ¼ 0 or, equivalently, A-1). In the no screen case the added mass coefficient goes from þ 1 to 1 as the oscillation frequency crosses the first or third sloshing frequency. The same feature is observed in the solid screen case at the second sloshing frequency. It is striking again that for the added mass coefficient, except for these two cases of solid or transparent screen, all the curves seem to intersect at the frequencies of the first and second sloshing modes. As for the free surface elevation RAOs, close inspection reveals that they do not actually intersect at the same points. As for the damping, it can be seen that resonance at the third sloshing frequency is little effective in dissipating energy: the damping coefficient Cb22 remains small. Much larger values are attained at the first two peaks. It is remarkable that, for a sway amplitude around 8 mm, the damping coefficient remains larger than 1 over a rather wide range of frequencies, from about 5 rad/s to about 9 rad/s. Harmonic Sway motion A=10mm ω=6 rad/s

Harmonic Sway motion A=10mm ω=8.5 rad/s

300

300

200

200 100

Fy in N

Fy in N

100 0 −100

0 −100

−200

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−300 54

54.5

55

55.5

56

Time in sec

56.5

57

57.5

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60

Time in sec

Fig. 10. Harmonic sway motion. Time series of the measured sway force for a motion amplitude of 10 mm and a frequency of 6 rad/s (left) or 8.5 rad/s (right). Thin full line: measured, thick full line: filtered, dash line: first-harmonic component.

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

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Figs. 8 and 9 show the measured and calculated added mass and damping coefficients Ca22, Cb22, vs. the oscillation frequency for sway amplitudes ranging from 2 mm through 12 mm. In the 2 mm and 6 mm cases, two series of experimental results appear: those tests were duplicated at several weeks interval (with the tank having been removed and put back in place again). Good agreement is observed between measured and calculated values up to motion amplitude of 10 mm where discrepancies start appearing around the second sloshing frequency. Alike for the free surface RAOs, these differences are associated with occurrence of breaking of the free surface.

A=1mm

2.5 2

2

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9 −1

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8

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10

11

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3

4

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6

7

8

ω rads−1

Fig. 11. Harmonic sway motion. Measured and calculated added mass coefficient Ca42 vs. frequency ω. From top to bottom, and left to right: motion amplitudes of 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm.

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B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

To illustrate the content of higher-order harmonics, we present, in Fig. 10, the time series of the measured force in two cases of relatively large sway amplitude (10 mm), at frequencies of 6 rad/s (left) and 8.5 rad/s (right). From Figs. 8 and 9, in the latter case, close to the second mode natural frequency, breaking of the free surface occurs and the measured hydrodynamic coefficients are somewhat off the calculated ones. The raw time series presents high frequency oscillations, around 10 Hz, which are due to vibrations of the Hexapode. The raw signal is filtered with a Butterworth filter which induces no phase shift. In both figures it can be checked that the filtered signal is very close to its first-harmonic component, meaning that higher-harmonic components are negligible. A=1mm

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A=8mm

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Fig. 12. Harmonic sway motion. Measured and calculated damping coefficient Ca42 vs. frequency ω. From top to bottom, and left to right: motion amplitudes of 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm.

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

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The following Figs. 11 and 12 show the off-diagonal coefficients Ca42 and Cb42, that is the roll hydrodynamic moment due to sway motion. The moment is expressed with respect to the mid bottom point (y¼0, z ¼ h) and made non-dimensional 2 3 with ρðbh þ 2b =3Þ. Good agreement is found between measured and calculated coefficients except around the second sloshing frequency at motion amplitudes larger than 10 mm. 4.2. Forced sway. Irregular imposed motion Figs. 13 and 14 show the added mass and damping coefficients Ca22 and Cb22, from measurements and calculations, for an imposed irregular sway motion of the tank. The experimental sway spectra have constant densities from 4 rad/s through 12 rad/s. In the calculations stochastic linearization is applied to the quadratic discharge equation, as described in Section 3.3. Cross-spectral analysis is applied to the measured hydrodynamic load to identify added mass and damping coefficients. The tests were run with 4 different values of the sway standard deviation, that is 1 mm, 2 mm, 4 mm and 8 mm. Again good agreement is obtained at the lower three values while, again due to breaking of the free surface, discrepancies appear around the second sloshing frequency at 8 mm. At any rate it appears that stochastic linearization of the discharge equation leads to satisfactory results. 4.3. Forced roll. Harmonic motion. Finally Figs. 15–18 show the hydrodynamic coefficients derived from the tests in forced roll. They are referred to the rollinduced sway amplitude at free surface level, for ease of comparison with the pure sway tests. Note that these roll-induced 2.5

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frequency (rad/s) Fig. 13. Irregular sway motion. Measured and calculated added mass coefficient Ca22.

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frequency (rad/s) Fig. 14. Irregular sway motion. Measured and calculated damping coefficient Cb22.

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B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

A=0.89mm

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A=1.8mm

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ω rads−1 Fig. 15. Harmonic roll motion. Measured and calculated added mass coefficient Ca24 vs. frequency sway amplitudes (at free surface level) of 0.9 mm, 1.8 mm, 2.4 mm, 4.8 mm, 6.4 mm, 9.6 mm.

7

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9

ω rads−1 ω. From top to bottom, and left to right: roll induced

sway amplitudes range from 1 mm to 10 mm, hence are somewhat lower than in the pure sway tests. As a result no breaking of the free surface was observed in the experiments, and very good agreement is reported between experimental and numerical values, for all 4 hydrodynamic coefficients, all over the frequency and amplitude ranges.

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

A=0.89mm

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Fig. 16. Harmonic roll motion. Measured and calculated damping coefficient Cb24 vs. frequency ω. From top to bottom, and left to right: roll induced sway amplitudes (at free surface level) of 0.9 mm, 1.8 mm, 2.4 mm, 4.8 mm, 6.4 mm, 9.6 mm.

5. Final comments An extensive series of experiments has been reported on the sloshing motion inside a tank fitted with a perforated screen, in forced sway or roll motion. The hydrodynamic loads, as measured by force sensors fitted to the test bench, have been expressed as added mass and damping coefficients, including coupling terms between sway and roll. Comparisons have been made with numerical results from a simple model based on linearized potential flow theory and quadratic

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B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

A=0.89mm

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Fig. 17. Harmonic roll motion. Measured and calculated added inertia coefficient Ca44 vs. frequency sway amplitudes (at free surface level) of 0.9 mm, 1.8 mm, 2.4 mm, 4.8 mm, 6.4 mm, 9.6 mm.

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pressure loss at the screen. Good agreement has been found in all cases where the free surface did not break. This result confirms that free surface nonlinearities less affect the hydrodynamic coefficients than they do the free surface elevations (Love and Tait, 2010). Damping coefficients have been found to be large over a wide frequency range, encompassing the first two sloshing frequencies. This is a remarkable result since it means that TLDs need not be accurately tuned and can be effective over a wide range of excitation frequencies.

B. Molin, F. Remy / Journal of Fluids and Structures 43 (2013) 463–480

A=0.89mm

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Fig. 18. Harmonic roll motion. Measured and calculated damping coefficient Cb44 vs. frequency ω. From top to bottom, and left to right: roll induced sway amplitudes (at free surface level) of 0.9 mm, 1.8 mm, 2.4 mm, 4.8 mm, 6.4 mm, 9.6 mm.

Experimental and numerical results have been presented in the case of irregular imposed sway motion, with good agreement. Considered extensions of the proposed numerical model are combined sway and roll motions, regular or irregular, and coupling with an exterior spring mass system alike in Love and Tait (2010, 2012) or Crowley and Porter (2012b). The case of multiple vertical screens can also easily be tackled.

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References Crowley, S., Porter, R., 2012a. The effect of slatted screens on waves. Journal of Engineering Mathematics 76, 53–76. Crowley, S., Porter, R., 2012b. An analysis of screen arrangements for a tuned liquid damper. Journal of Fluids and Structures 34, 291–309. Faltinsen, O., Firoozkoohi, R., Timokha, A.N., 2011. Steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle: quasilinear modal analysis and experiments. Physics of Fluids 23, 042101. Faltinsen, O., Timokha, A.N., 2001. An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. Journal of Fluid Mechanics 432, 167–200. Kimmoun, O., Molin, B., Moubayed, W., 2001. Second-order analysis of the interaction of a regular wave train with a vertical perforated wall. In: Actes 8èmes Journées de l'Hydrodynamique, Nantes, pp. 85–98. http://website.ec-nantes.fr/actesjh/ (in French). Love, J.S., Tait, M.J., 2010. Nonlinear simulation of a tuned liquid damper with damping screens using a modal expansion technique. Journal of Fluids and Structures 26, 1058–1077. Love, J.S., Tait, M.J., 2012. A preliminary design method for tuned liquid dampers conforming to space restrictions. J. Fluids Struct. 40, 187–197. Molin, B., 2011. Hydrodynamic modeling of perforated structures. Applied Ocean Research 33, 1–11. Molin, B., Fourest, J.-M., 1992. Numerical modeling of progressive wave absorbers. In: Proceedings 7th International Workshop on Water Waves and Floating Bodies, Val de Reuil. www.iwwwfb.org. Molin, B., Legras, J.-L., 1990. Hydrodynamic modeling of the Roseau tower stabilizer. In: Proceedings 9th OMAE Conference, Houston, pp. 329–336. Tait, M.J., 2008. Modelling and preliminary design of a structure-TLD system. Engineering Structures 30, 2644–2655. Taylor, G.I., 1953. An experimental study of standing waves. Proceedings Royal Society of London A 218, 44–59. Warnitchai, P., Pinkaew, T., 1998. Modelling of liquid sloshing in rectangular tanks with flow-dampening devices. Engineering Structures 20, 593–600.