Second-order resonance of sloshing in a tank

ARTICLE IN PRESS

Ocean Engineering 34 (2007) 2345–2349 www.elsevier.com/locate/oceaneng

Second-order resonance of sloshing in a tank G.X. Wu1 Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK Received 5 February 2007; accepted 15 May 2007 Available online 21 May 2007

Abstract Sloshing in a two-dimensional rectangular tank in horizontal motion is analysed based on the velocity potential theory. It is found that even when the first-order excitation is away from all the natural frequencies of the tank, second-order resonance can still occur when the sum-frequency or the difference-frequency is equal to one of the natural frequencies corresponding to the even mode. However, such resonance is not excited when the sum or difference frequency is equal to the natural frequency of an odd mode. r 2007 Elsevier Ltd. All rights reserved. Keywords: Second-order resonance; Sloshing; Sum and difference frequency

1. Introduction It is well known that many systems in oscillation have resonant frequencies. One of such cases in marine engineering is sloshing in a tank. Based on the linear velocity potential theory, the resonant frequencies corresponding to a rectangular tank can be easily established (e.g., Faltinsen, 1978). When the tank is set into periodical motion at one of those resonant frequencies, the linear theory shows that the elevation of the liquid surface may tend to infinity as time progresses. When the fully nonlinear theory is used (e.g., Wu et al., 1998; Faltinsen and Timokha, 2001; Wang and Khoo, 2005), the surface elevation may not go to infinity, but it remains extremely large. Wave overturning and breaking may occur. One interesting issue is the second-order wave elevation. When the first-order wave has components of frequencies on, n ¼ 0, 1, 2, y, the second-order wave will have components of oi7oj, i,j ¼ 0, 1, 2, y. One would then obviously speculate that even if none of the excitation frequencies On is a natural frequency, second-order resonance may occur if Oi7Oj is. Indeed, numerical E-mail address: [email protected] Cheung Kong and Long Jiang visiting professor, College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China. 1

0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.05.004

evidence seems to confirm such an intuitive hypothesis in the problem of a group of cylinders near the trapped mode (Malenica et al., 1999; Wang and Wu, 2007). It was found that when the wave frequency was half of that at the trapped mode, the first-order result did not show any abnormality, but the second-order results became large because the double frequency was at the trapped mode. However, for the problem of a tank in sway oscillation at half of its first natural frequency, nothing particular in the second-order results was observed in the numerical analysis. The discrepancy between these two cases has led to the detailed analysis for sloshing in this paper. It is then found that in sway motion when Oi7Oj is equal to any odd mode of the natural frequencies, no second-order resonance will occur. When it is equal to one of the even modes, second-order resonance does occur. 2. Mathematical analysis We consider a two-dimensional rectangular tank of depth d and length 2l undergoing horizontal oscillation with displacement s(t). A Cartesian coordinate system ox0y0 fixed in the space is defined, with the origin on the mean free surface and in the middle of the tank when it is stationary, and y0 pointing upwards. When the fluid is inviscid and the flow is irrotational, the fluid motion can be described by a velocity potential f which satisfies the

ARTICLE IN PRESS G.X. Wu / Ocean Engineering 34 (2007) 2345–2349

2346

Laplace equation

r 2f ¼ 0

(1)

in the fluid domain R(t). The kinematic and dynamic conditions on the free surface SF or y ¼ Z can be written as qf qZ qf qZ   ¼ 0, qy0 qt qx0 qx0

(2)

qf 1 þ gZ þ r fr f ¼ 0, (3) qt 2 where g is the acceleration due to gravity. On the tank surface, we have qf ¼ UðtÞ; x0 ¼ l þ sðtÞ, qx0 qf ¼ 0; y0 ¼ d, qy0

qfð2Þ q2 fð1Þ qfð1Þ þ gZð2Þ ¼  Zð1Þ þU qt qyqt qx 1  r fð1Þ r fð1Þ . ð17Þ 2 Similarly, substituting Eqs. (12) and (13) into (10) and (11), we obtain qfð1Þ ¼ UðtÞ; qx qfð1Þ ¼ 0; qy

qfð2Þ ¼ 0; qx

qfð2Þ ¼ 0; qy

x ¼ l,

(18)

y ¼ d.

(19)

The solution of f(1) can be written as (4) fð1Þ ¼ Ux þ (5)

1 X

An ðtÞ

n¼1

cosh kn ðy þ dÞ cos kn ðx þ lÞ, cosh kn d

(20)

where UðtÞ ¼ s_ðtÞ. The initial conditions can be written as

where kn ¼ np=2l. This expansion satisfies all the equations apart from the boundary conditions in (14) and (15), or

Zðx0 ; 0Þ ¼ 0; fðx0 ; 0; 0Þ ¼ 0.

g

(6)

We define another Cartesian system oxy which is fixed on the tank. The relationships between these two systems can be written as x ¼ x0  sðtÞ; y ¼ y0 .

(7)

qfð1Þ q2 fð1Þ þ ¼ 0. qy qt2

(21)

Substituting Eq. (20) into (21), we obtain 1 X € ½An ðtÞgkn tanh kn d þ A€ n ðtÞ cos kn ðx þ lÞ ¼ Ux. n¼0

Eq. (1) then has the same form in oxy. Eqs. (2) and (3) become

Using the orthogonality of the cosine functions cos kn ðx þ lÞ over (l,l), we have

qf qZ qZ qf qZ  þU  ¼0 qy qt qx qx qx

(8)

ð1Þ  1 . An gkn tanh kn d þ A€ n ¼ U€ k2n l

qf qf 1 U þ gZ þ r fr f ¼ 0, qt qx 2

(9)

The initial condition of An(t) can be obtained from Eq. (6), or

qf ¼ UðtÞ; x ¼ l, qx

(10)

qf ¼ 0; qy

(11)

y ¼ d.

pffiffiffiffiffiffi We now assume that  ¼ U= gd is a small parameter. This allows us to use the standard perturbation theory to write f ¼ f

ð1Þ

2

þ f

ð2Þ

þ ...,

f ¼ Zð1Þ þ 2 Zð2Þ þ . . . .

n

Uð0Þx þ _ Uð0Þx þ

1 X n¼1 1 X

Substituting these two equations into (8) and (9) and expanding the results from y ¼ Z to y ¼ 0, we have qfð1Þ qZð1Þ  ¼ 0, qy qt

(14)

qfð1Þ þ gZð1Þ ¼ 0, qt

(15)

qfð2Þ qZð2Þ qZð1Þ qfð1Þ qZð1Þ q2 fð1Þ  ¼ U þ  Zð1Þ , qy qt qx qx qx qy2

(16)

Að0Þ cos kn ðx þ lÞ ¼ 0, _ Að0Þ cos kn ðx þ lÞ ¼ 0.

ð23Þ

n¼1

This gives Að0Þ ¼ Uð0Þ

(12) (13)

(22)

ð1Þn  1 ; k2n l

n

ð1Þ  1 _ _ Að0Þ ¼ Uð0Þ . k2n l

(24)

The solution of Eq. (22) can be easily obtained as Z t 2 ð1Þn  1 d U sin on ðt  tÞ dt An ¼ 2 2 on k n l 0 dt  _ þon Uð0Þ cos on t þ Uð0Þ sin on t   Z t ð1Þn  1 ¼ UðtÞ sin on ðt  tÞ dt , ð25Þ UðtÞ  on k2n l 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where on ¼ gkn tanh kn d . Eq. (20) then becomes fð1Þ ¼

1 X n¼1

Bn ðtÞ

cosh kn ðy þ dÞ cos kn ðx þ lÞ, cosh kn d

(26)

ARTICLE IN PRESS G.X. Wu / Ocean Engineering 34 (2007) 2345–2349

where Bn ¼ 

ð1Þn  1 on k2n l

Z

t

UðtÞ sin on ðt  tÞ dt.

(27)

0

From Eq. (15), we also have Zð1Þ ¼ 

ð1 þ tanh kn d tanh km dÞ 1  cos½ðkn þ km Þðx þ lÞ þ U U_ 2 1 X ¼ q0 ðtÞ þ qn ðtÞ cos kn ðx þ lÞ,

1 1X B_ n ðtÞ cos kn ðx þ lÞ. g n¼1

(28)

where q0 ðtÞ ¼

 A_ n An k2n ð1 þ tanh2 kn dÞ, qn ðtÞ ¼

(29)

!

q q2 fð1Þ qfð1Þ 1 Zð1Þ þ þU  r fð1Þ r fð1Þ qx 2 qyqt qx 1 X 1 X

!

þ

1 1 X ½B_ m Amn km kmn þ B_ m Bmn k2mn 2 m¼nþ1

þ ðAm B_ mþn km tanh km d þ Amþn B_ m kmþn  tanh kmþn dÞkmþn tanh kmþn d 1  ðA_ m Amþn þ Am A_ mþn Þkm kmþn 2 Hðn  2Þ ð1 þ tanh km d tanh kmþn dÞ þ 2 n1 X  ½B_ m Anm kn knm þ B_ m Bnm k2nm

B_ n Am kn km sin kn ðx þ lÞ sin km ðx þ lÞ

n¼1 m¼1 1 X 1 X

ð31Þ

 tanh kmn dÞkmn tanh kmn d 1  ðA_ m Amn þ Am A_ mn Þkm kmn 2 ð1 þ tanh km d tanh kmn dÞ 1 1X ½B_ m Amþn km kmþn þ B_ m Bmþn k2mþn þ 2 m¼1

where

¼ 

1 1 _ X UU þ ½B_ n Bn k2n þ An B_ n k2n ð2 tanh2 kn d  1Þ 2 n¼1

þ ðAm B_ mn km tanh km d þ Amn B_ m kmn

qfð2Þ q2 fð2Þ þ ¼ qðx; tÞ, qy qt2

qZð1Þ qfð1Þ qZð1Þ q2 fð1Þ þ  Zð1Þ qðx; tÞ ¼ g U qx qx qx qy2

ð30Þ

n¼1

Eq. (27) shows that when U(t) contains a periodic component of on, n ¼ 1, 3, y, the integration gives a term of t. Thus Bn tends to infinity as t-N and resonance occurs. For the second-order potential, we have from Eqs. (16) and (17) g

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B_ n Bm k2m cos kn ðx þ lÞ cos km ðx þ lÞ

n¼1 m¼1 1 X 1 1X þ ðB€ n B_ m þ B_ n B€ m Þkm tanh km d g n¼1 m¼1

m¼1

þ ðAm B_ nm km tanh km d þ Anm B_ m knm  tanh knm dÞknm tanh knm d 1  ðA_ m Anm þ Am A_ nm Þkm knm 2 ð1 þ tanh km d tanh knm dÞ.

 cos kn ðx þ lÞ cos km ðx þ lÞ 1 X 1 1 1X ðA_ n Am þ An A_ m Þ þ U U_  2 2 n¼1 m¼1 kn km ½sin kn ðx þ lÞ sin km ðx þ lÞ þ tanh kn d tanh km d cos kn ðx þ lÞ  cos km ðx þ lÞ. Noticing B€ n ¼ o2n An ¼ gkn tanh kn d, we have 1 X 1 1X ½B_ n Am kn km þ B_ n Bm k2m qðx; tÞ ¼ 2 n¼1 m¼1

þ ðAn B_ m kn tanh kn d þ Am B_ n km tanh km dÞ 1 km tanh km d  ðA_ n Am þ An A_ m Þkn km 2 ð1 þ tanh kn d tanh km dÞ cos½ðkn  km Þðx þ lÞ 1 X 1 1X ½B_ n Am kn km þ B_ n Bm k2m þ 2 n¼1 m¼1 þ ðAn B_ m kn tanh kn d þ Am B_ n km tanh km dÞ 1 km tanh km d  ðA_ n Am þ An A_ m Þkn km 2

ð32Þ

Here H(n) ¼ 0, no0 and H(n) ¼ 1, nX0. The solution of the second-order potential can then be written as fð2Þ ¼ C 0 ðtÞ þ

1 X n¼1

C n ðtÞ

cosh kn ðy þ dÞ cos kn ðx þ lÞ, cosh kn d

(33)

where 1 C 0 ðtÞ ¼ 2

Z t( 0

" 1 1 2 X 1 2 U þ B2n k2n þ 2 B_ n k2n 2 on n¼1 2

ð2 tanh kn d  1Þ 

A2n k2n ð1

#) 2

þ tanh kn dÞ

dt, ð34Þ

Z C n ðtÞ ¼

t

qn ðtÞ sin on ðt  tÞdt. 0

(35)

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Similar to Eq. (27), when there is a term of on in qn(t), Cn(t)-N as t-N and resonance can occur.

The result shows that the first-order resonance occurs when O0 ¼ on , n ¼ 1; 3; . . ., as discussed previously. It ought to point out that there are no even terms, or

A2n ¼ B2n ¼ 0, n ¼ 1; 2; . . .. As a result, Eq. (32) gives q2n1 ¼ 0, n ¼ 1; 2; . . .. In other words, while the first potential has only the odd terms, the secondorder potential has only the even terms. This is why when 2O0 ¼ o1 which corresponds to an odd mode, no second-order resonance is observed. Further inspection of Eq. (32) shows that q2n contains components of 2O0, |O07o2m1|, omþ2n  om ðm ¼ 1; 3; :::Þ, om þ o2nm ðm ¼ 1; . . . ; 2n  1Þ. Obviously, if any of these components is equal to one of the even modes, second-order resonance can occur. pffiffiffiffiffiffiffiffi 2 pFig. ffiffiffiffiffiffiffiffi 1a and 1b give f ¼ C 2 =ðU 0 d=gÞ against s ¼ t g=d for l ¼ d, and at 2O0 ¼ o1 and 2O0 ¼ o2 , respectively. The former case corresponds to an odd mode and no resonance occurs. The latter case corresponds to an even mode and the resonance is obvious. Fig. 2a and 2b give the same result at O0 þ o1 ¼ o2 and O0  o1 ¼ o2 , respectively. The resonance occurs in both cases.

Fig. 1. (a) C2 against t at 2O0 ¼ o1. (b) C2 against t at 2O0 ¼ o2.

Fig. 2. (a) C2 against t at O0+o1 ¼ o2. (b) C2 against t at O0o1 ¼ o2.

3. Numerical results We first consider an example with UðtÞ ¼ U 0 sin O0 t,

(36)

we have Bn ¼ on U 0

An ¼ O0 U 0

ð1Þn  1 on sin O0 t  O0 sin on t , o2n  O20 k2n l

ð1Þn  1 O0 sin O0 t  on sin on t . o2n  O20 k2n l

(37)

(38)

ARTICLE IN PRESS G.X. Wu / Ocean Engineering 34 (2007) 2345–2349

An ¼

J X

Oj U j

j¼0

ð1Þn  1 Oj sin Oj t  on sin on t . o2n  O2j k2n l

2349

(40)

Following the previous discussions, we can establish that when Oj  o2m1 ¼ o2n or Oi  Oj ¼ o2n , i; j ¼ 0; 1; . . . J, the second-order resonance can occur. The first case is virtually identical to that shown in Fig. 2a and b. For the second case, Fig. 3a gives results for J ¼ 1 and U 0 ¼ U 1 at O0 ¼ o2 =4 and O1 ¼ 3o2 =4, while Fig. 3b corresponds to O0 ¼ o2 =4 and O1 ¼ 5o2 =4. As expected, resonance occurs in both cases.

4. Conclusions

Fig. 3. (a) C2 against t at O0 ¼ o2/4 and O1 ¼ 3o2/4. (b) C2 against t at O0 ¼ o2/4 and O1 ¼ 5o2/4.

Based on the discussion above, the second-order resonance may occur when omþ2n  om ¼ o2n . But this equation can be satisfied only for a particular value of d/(2l). In fact it is possible only when d/(2l) is small. If we consider an example with o3  o1 ¼ o2 and use pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on ¼ gkn tanh kn d , we find d=ð2lÞ ¼ 3:75  103 , which is extremely shallow water. Thus any possible second-order resonance under this condition may be of limited practical interest. We next consider a case with UðtÞ ¼

J X

U j sin Oj t.

j¼0

This gives Bn ¼ 

J X j¼0

on U j

ð1Þn  1 on sin Oj t  Oj sin on t , o2n  O2j k2n l

(39)

In addition to the first-order resonance when the excitation is at one of the natural frequencies on, n ¼ 1, 2, 3, y, the second-order resonance can occur when the sum frequency or the difference frequency of any two excitation components is equal to one of the natural frequencies. The second-order resonance can also occur when the sum (or the difference) of any one of its excitation frequencies and any one of the natural frequencies is equal to another natural frequency. Exception can occur when some modes are not part of the solution of the problem, i.e., when the corresponding coefficient in the expansion in Eq. (33) is zero. It ought to be emphasized that these conclusions are a result of the perturbation theory. The second-order resonance occurs because the first-order results are the separate forcing terms in the analysis. When the fully nonlinear theory is used, the forcing terms and responding terms are coupled. The second-order resonant behaviour could become less obvious. Thus any possible significance of the present paper may be only in the context of the perturbation theory.

References Faltinsen, O.M., 1978. A numerical non-linear method for sloshing in tanks with two dimensional flow. Journal of Ship Research 18 (4), 224–241. Faltinsen, O.M., Timokha, A.N., 2001. An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. Journal of Fluid Mechanics 432, 167–200. Malenica, S., Eatock Taylor, R., Huang, J.B., 1999. Second order water wave diffraction by an array of vertical cylinders. Journal of Fluid Mechanics 390, 349–373. Wang, C.Z., Khoo, B.C., 2005. Finite element analysis of two-dimensional nonlinear sloshing problems in random excitation. Ocean Engineering 32, 107–133. Wang, C.Z., Wu, G.X., 2007. Time domain analysis of second-order wave diffraction by an array of vertical cylinders. Journal of Fluids Structures 23, 605–631. Wu, G.X., Ma, Q.W., Eatock Taylor, R., 1998. Numerical simulation of sloshing waves in a 3D tank based on a finite element method. Applied Ocean Research 20, 337–355.