Liquid sloshing in rigid cylindrical container with multiple rigid annular baffles: Free vibration

Journal of Fluids and Structures 34 (2012) 138–156

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Liquid sloshing in rigid cylindrical container with multiple rigid annular baffles: Free vibration J.D. Wang a, S.H. Lo b, D. Zhou a,n a b

College of Civil Engineering, Nanjing University of Technology, Nanjing 210009, China Department of Civil Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, China

a r t i c l e in f o

abstract

Article history: Received 22 August 2011 Accepted 18 June 2012 Available online 19 July 2012

A semi-analytical approach is presented to obtain the natural frequencies and vibration modes of ideal liquid sloshing in a rigid partially liquid-filled cylindrical container with multiple rigid annular baffles of the same inner radius. The complicated liquid domain is divided into several simple sub-domains so that the liquid velocity potential in each liquid sub-domain is of class C1 with continuity boundary conditions. Based on the superposition principle, the analytical solutions of the liquid velocity potential corresponding to each liquid sub-domain are obtained by means of the method of separation of variables. The eigenfrequency equation is obtained by expanding the free surface condition and the artificial interface conditions into the Fourier series in the liquid height direction and the Bessel series in the radial direction. Stable and fast numerical computations are observed by the convergence study. Excellent agreements have been achieved in the comparison of results obtained by the proposed approach with those given by finite element software ADINA. The natural frequencies and mode shapes versus the position, the inner radius and the number of the annular baffles are discussed in detail. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Cylindrical container Multiple baffles Sub-domains Sloshing Analytical solution

1. Introduction Sloshing is one kind of liquid motion in a container with a free surface, which is caused by a disturbance to the container with partially filled liquid. The liquid sloshing problems in moving or stationary containers remain of great concern to the dynamic safety of liquid-structure coupling systems, such as space vehicles, road vehicle tanks, oil ships, storage tanks and elevated water towers under ground motion. Liquid sloshing in a rigid container has been studied for many years. Abramson (1966) presented the analytical solutions of liquid sloshing in containers with simple shapes such as rectangular containers and vertical circular (or annular) cylindrical containers. Mciver (1989) investigated 2-D liquid sloshing in a horizontal circular cylindrical container and 3-D liquid sloshing in a spherical container. Hasheminejad and Aghabeigi (2009) studied the sloshing frequencies of transverse modes in a half-filled rigid horizontal cylindrical container with elliptical cross-section. Marsh et al. (2011) used the smoothed particle hydrodynamics to model the free surface wave in a liquid sloshing absorber which consists of a rigid container partially filled with liquid.

n

Corresponding author. E-mail address: [email protected] (D. Zhou).

0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2012.06.006

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

139

It is important to understand the coupled vibration characteristics of liquid–container interaction where the container flexibility should be considered. Amabili (1996) investigated the coupled vibration of the horizontal flexible cylindrical shell partially filled with liquid. Amabili (1997) studied the bulging modes of a flexible bottom in circular and annular cylindrical containers partially filled with liquid. Amabili et al. (1998) analyzed the dynamic characteristics of the vertical flexible cylindrical container partially filled with liquid. Furthermore, Amabili (2000) studied the vibrations of plates, shells and plate-shell systems coupled with the quiescent and inviscid liquid where the liquid sloshing was included. Zhou and Liu (2007) studied the free vibration of elastic rectangular tanks partially filled with liquid by using a combination of analytical method and Ritz method. Curadelli et al. (2010) investigated the dynamical response of elevated elastic spherical container partially filled with liquid subjected to horizontal base motion. To suppress liquid sloshing, baffles are generally installed in liquid storage containers. It is well known that using such a baffle will result in energy dissipation which could reduce the liquid sloshing amplitude and the hydrodynamic load. At the same time, the baffles also improve the stiffness of the containers. Evans and McIver (1987) studied the effect of ¨ introducing a vertical thin baffle into a rectangular water tank on the sloshing frequencies. Gedikli and Erguven (1999) analyzed the seismic response of a rigid liquid-filled container with a rigid baffle using the boundary element method. Biswal et al. (2003) studied the natural frequencies and modes of liquid in a liquid-filled cylindrical rigid container with/ without baffles by using the finite element method. Cho et al. (2005) numerically calculated the resonance characteristics of liquid sloshing in a 2-D baffled container subjected to lateral excitation. Gavrilyuk et al. (2006) used the Green function method to obtain frequencies and modes of the liquid sloshing in a rigid vertical circular cylindrical container with a rigidring horizontal thin baffle. Maleki and Ziyaeifar (2007) investigated the damping effect of baffles on the seismic response of isolated cylindrical liquid storage containers. Firouz-Abadi et al. (2009) used the boundary element method to determine the frequencies and mode shapes of liquid sloshing in 3-D baffled containers with arbitrary geometries. Goudarzi et al. (2010) developed an analytical model to estimate the hydrodynamic damping ratio of liquid sloshing for wall bounded baffles. Recently, Wang et al. (2010) investigated the liquid sloshing in a rigid cylindrical container with a rigid annular baffle using the analytical method. Askari and Daneshmand (2010) and Askari et al. (2011) investigated the effects of a rigid internal body on dynamic characteristics of a cylindrical container partially filled with liquid. Hasheminejad and Aghabeigi (2012) investigated the effects of surface-piercing or bottom-mounted vertical baffles on sloshing characteristics of liquid in a half-full rigid horizontal cylindrical container of elliptical cross section. It should be mentioned that multiple annular baffles have been used in launch vehicles and petroleum storages to suppress the liquid sloshing and improve the dynamic safety of the system. However, searching the available references, one can find that no paper studied the effect of multiple horizontal baffles on the sloshing characteristics of liquid in partially liquid-filled containers. In this paper, an improved semi-analytical solution is developed to study the frequencies and modes of liquid sloshing in a rigid cylindrical container with multiple rigid annular baffles of the same inner radius. A two-step partition is developed to divide the liquid domain into several sub-domains along the baffles. Based on the superposition principle, the liquid velocity potential corresponding to each liquid sub-domain can be obtained by means of the method of separation of variables. Fourier series and Bessel series expansions are used to match the free surface condition and the artificial interface conditions. Frequencies and modes with high accuracy have been obtained by solving the linear eigenvalue equation.

2. Physical descriptions and modeling Consider multiple thin rigid annular baffles placed in a rigid cylindrical container partially filled with inviscid, incompressible and irrotational liquid, as shown in Fig. 1. The movement of the liquid is described by the cylindrical coordinate system with its origin located at the center of the bottom of the container. The free surface of the liquid is

Fig. 1. The rigid partially liquid-filled cylindrical container with multiple rigid annular baffles.

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orthogonal to the z axis. The free surface waves are considered in the present analysis and a linear theory of liquid movement is adopted from the small-amplitude sloshing of the liquid. All the baffles have the same inner radius and their thickness is very small compared to the outer radius of the baffles, therefore the effect of the baffle thickness on liquid movement can be neglected. Let M be the total number of the baffles and z ¼h1, h2,..., hM, respectively, be the heights of these baffles. We use hM þ 1 to denote the liquid height and use h0 ¼0 to denote the position of the container bottom. The radius of the container is R2 and the inner radius of the baffles is R1. According to the above assumptions and definitions, the velocity potential of liquid movement should satisfy the Laplace equation:   1@ @f 1 @2 f @2 f r þ 2 2 þ 2 ¼ 0 in O, ð1Þ r @r @r @z r @y where O is the liquid domain. Velocity potential f and the velocity of the liquid are related by partial derivatives as follows: vr ¼

@f @f , vy ¼ , @r r@y

vz ¼

@f : @z

ð2Þ

The impermeability condition of the liquid at rigid boundaries is @f ¼0 @n

on

Gw ,

ð3Þ

where Gw is the wetted solid–liquid interfaces; n is the normal vector outward from the liquid domain O. The velocity potential should be finite at r ¼0, i.e.

f9r ¼ 0 ¼ finite value:

ð4Þ

It is obvious that f should be 2p-periodic along the circumference, i.e.

fðy þ2pÞ ¼ fðyÞ:

ð5Þ

Taking the effect of the surface waves into account, the motion equation of the free surface is @f 1 @2 f þ ¼0 g @t 2 @z

on

Gf ,

ð6Þ

where Gf is the free surface of the liquid. 3. Partition of liquid domain The analytical solutions of liquid sloshing in a simple container such as rectangular container and vertical cylindrical container can be determined directly by using the method of separation of variables (Abramson, 1966). However, this method is somewhat less powerful for the case of liquid in a baffled container because the liquid domain becomes complicated one due to the existence of baffles submerged in the liquid. In the present study, a two-step partition is developed in the solution process, as shown in Fig. 2. Firstly, the liquid domain is divided into cylindrical liquid domains between two adjacent baffles. Secondly, each cylindrical liquid domain is divided into a cylindrical liquid sub-domain and an annular cylindrical liquid sub-domain. By means of such a partition, the liquid velocity potential in each sub-domain is of class C1 with continuity boundary conditions. Therefore, the liquid domain with M baffles can be divided into 2Mþ2 sub-domains with 2Mþ1 artificial interfaces (i.e. Mþ1 cylindrical artificial interfaces and M circular artificial interfaces),

Fig. 2. Cross-section, sub-domains and artificial interfaces of the system.

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

141

namely

fðr, y,z,tÞ ¼ fi ðr, y,z,tÞ, ðr, y,zÞ 2 Oi

ði ¼ 1,. . .,2M þ 2Þ,

ð7Þ

in which, the naming rules of the liquid sub-domains and the artificial interfaces are as follows: 1. 2. 3. 4.

the the the the

Mþ 1 annular cylindrical liquid sub-domains from bottom to top are named as O1,O3,y,O2M þ 1; Mþ 1 cylindrical liquid sub-domains from bottom to top are named as O2,O4,y,O2M þ 2; Mþ 1 cylindrical artificial interfaces from bottom to top are named as G1,G2,y,GM þ 1; M circular artificial interfaces from the first baffle to the last baffle are named as GM þ 2,GM þ 3,y,G2M þ 1.

According to the above definitions, we have ( fðr, y,zÞ : R1 r r r R2 ,0 r y r 2p,hp r z r hp þ 1 g, ðp ¼ 0,1,. . .,M, i ¼ 2p þ 1Þ : Oi ¼ fðr, y,zÞ : 0 r r rR1 ,0 r y r 2p,hp r z r hp þ 1 g, ðp ¼ 0,1,. . .,M, i ¼ 2p þ 2Þ 0

ð8Þ 0

If Oi and Oi0 (ioi ) contact each other, then their artificial interface should be Gk and the ordered triple (i, i , k) should satisfy: ði, i0 , kÞ 2 fð2p1,2p,pÞ9p ¼ 1,2,. . .,M þ 1g [ fð2p,2p þ 2,M þ 1 þpÞ9 p ¼ 1,. . .,Mg:

ð9Þ

Obviously, Oi and Oi satisfy pressure continuity condition and velocity continuity condition on the artificial interfaces, 0

i.e. @fi0 @fi @fi @fi0 ¼ , ¼ ,on Gk : @t @t @nk @nk in which, nk is the normal vector to Gk. The impermeability conditions at rigid boundaries are given by    @f1  @f  @f  ¼ 0, 2  ¼ 0, i  ¼0 ðp ¼ 0,1,. . .,M,i ¼ 2p þ 1Þ,  @z z ¼ 0 @z z ¼ 0 @r r ¼ R2   @fi  @f 0  ¼ 0, i  ¼0  @z z ¼ hp @z z ¼ hp

ðp ¼ 1,. . .,M, i ¼ 2p1, i0 ¼ 2p þ1Þ:

ð10Þ

ð11Þ

ð12Þ

It should be noted that O2M þ 1 and O2M þ 2 have free surfaces G2M þ 3 and G2M þ 2, respectively. In such a case, Eq. (6) becomes     @fi  1 @2 fi  þ ¼ 0 on Gk ,ði,kÞ 2 ð2M þ1,2M þ3Þ,ð2M þ 2,2M þ 2Þ : ð13Þ  @z z ¼ h g @t 2  z¼h

4. Solution of velocity potential When liquid is undergoing free sloshing in the container, f is the harmonic functions of time, therefore

fðr, y,z,tÞ ¼ joejot Fðr, y,zÞ,

ð14Þ

in which, o is the natural frequency of liquid sloshing. j is the imaginary unit. F is the mode shape of the liquid domain O and Fi (i ¼1,y,2Mþ2) is that within the sub-domain Oi. Substituting Eq. (14) into Eqs. (11)–(12) yields    @F1  @F2  @Fi  ¼ 0, ¼ 0, ¼ 0, ðp ¼ 0,1,. . .,M, i ¼ 2p þ 1Þ, ð15Þ @z z ¼ 0 @z z ¼ 0 @r r ¼ R2   @Fi  @Fi0  ¼ 0, ¼ 0, ðp ¼ 1,. . .,M,i ¼ 2p1, i0 ¼ 2p þ 1Þ: @z z ¼ hp @z z ¼ hp

ð16Þ

According to Fig. 2 and Eqs. (15)–(16), the mode shape within liquid sub-domain Oi, upon using the principle of superposition, can be written as 8 > Qi < 1, ðp ¼ 1,. . .,M,i ¼ 2p1Þ X q ði ¼ 2,2M þ 1Þ ð17Þ Fi ¼ Fi , Q i ¼ 2, , > : 3, ðp ¼ 1,. . .,M,i ¼ 2p þ2Þ q¼1 in which, Qi denotes the total number of the free surface wave equation and interface conditions, Fqi denotes the qth kind velocity potential function. For annular cylindrical liquid sub-domain Oi (p ¼0,1,y,M  1, i¼2pþ1), F1i satisfies the

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corresponding boundary conditions given as follows:    @F1i  @F1i  @F1i  ¼ 0, ¼ 0, ¼ 0:    @z  @z  @r  z ¼ hp þ 1

z ¼ hp

ð18Þ

r ¼ R2

Namely, F1i satisfies the rigid boundary conditions on the upper and lower surfaces as well as on the outer lateral surface. For annular cylindrical liquid sub-domain O2M þ 1, Fq2M þ 1 (q¼1,2) satisfies the corresponding boundary conditions given as follows:    @F12M þ 1  @F12M þ 1   F12M þ 1  ¼ 0, ¼ 0, ¼0   z ¼ hM þ 1 @z  @r  z ¼ hM r ¼ R2    @F22M þ 1  @F22M þ 1  @F22M þ 1  ¼ 0, ¼ 0, ¼ 0: ð19Þ    @r  @r  @z  r ¼ R2

r ¼ R1

z ¼ hM

F12M þ 1

satisfies the zero pressure condition on the upper surface and the rigid boundary conditions on the Namely, outer lateral and lower surfaces. F22M þ 1 satisfies the rigid boundary conditions on outer lateral and inner lateral surfaces as well as on the lower surface. For cylindrical liquid sub-domain O2, Fq2 (q¼1,2) satisfies the corresponding boundary conditions given as follows:    @F12  @F12   ¼ 0, ¼ 0, F12  ¼ finite value,   @z  @z  r¼0 z ¼ h1 z ¼ h0    @F22  @F22   ¼ 0, F22  ¼ finite value, ¼ 0: ð20Þ   @r  r¼0 @z  r ¼ R1

z ¼ h0

Namely, F12 satisfies the rigid boundary conditions on the upper and lower surfaces; F22 satisfies the rigid boundary conditions on the lateral and lower surfaces. For cylindrical liquid sub-domain Oi (p¼1,y,M  1, i¼2p þ2), Fqi (q¼1,2,3) satisfies the corresponding boundary conditions given as follows:    @F1i  @F1i   ¼ 0, ¼ 0, F1i  ¼ finite value,   @z  @z  r¼0 z ¼ hp þ 1 z ¼ hp    @F2i  @F2i   ¼ 0, F2i  ¼ finite value, ¼ 0,   r¼0 @r  @z  r ¼ R1 z ¼ hp    @F2i  @F3i  3 ¼ 0, Fi  ¼ finite value, ¼ 0: ð21Þ   r¼0 @z  @z  z ¼ hp

z ¼ hp þ 1

F1i

satisfies the rigid boundary conditions on the upper and lower surfaces. F2i satisfies the rigid boundary Namely, conditions on the lateral and lower surfaces. F3i satisfies the rigid boundary conditions on the lateral and upper surfaces. For cylindrical liquid sub-domain O2M þ 2, Fq2M þ 2 (q¼ 1,2,3) satisfies the corresponding boundary conditions given as follows:    @F12M þ 2    F12M þ 2  ¼ 0, ¼ 0, F12M þ 2  ¼ finite value,  z ¼ hM þ 1 @z  r¼0 z ¼ hM    @F22M þ 2  @F22M þ 2   ¼ 0, F22M þ 2  ¼ finite value, ¼ 0,   @r  r¼0 @z  r ¼ R1 z ¼ hM    @F32M þ 2    ¼ 0, F32M þ 2  ¼ finite value, F32M þ 2  ¼ 0: ð22Þ  @r  r¼0 z ¼ hM þ 1 r ¼ R1

F12M þ 2

satisfies the zero pressure condition on the upper surface and the rigid boundary condition on the lower Namely, surface. F22M þ 2 satisfies the rigid boundary conditions on the lateral and lower surfaces. F32M þ 2 satisfies the rigid boundary conditions on the lateral and upper surfaces. It should be mentioned that in the above analysis, the general solutions of Fqi is obtained by using the method of separation of variables, according to Eqs. (17)–(22). Obviously, the governing equation (1) and impermeability condition (3) have been satisfied. However, the free surface wave equation was replaced by the zero pressure condition in Eqs. (19) and (22). In the following study, the undetermined coefficients in the general solutions will be determined by substituting Fqi into the free surface wave eq. (6) and the interface equation (10). For simplification in analysis, the following dimensionless coordinates and parameters are introduced: sffiffiffiffiffiffi h r z R n R x ¼ , z ¼ , L ¼ o 2 , gk ¼ k , a ¼ 1 , bp ¼ p , ðp ¼ 0,1,. . .,M þ1Þ: ð23Þ R2 R2 g R2 R2 R2

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

143

The mode shapes of liquid sloshing can be distinctly classified into the symmetric and anti-symmetric ones. Without loss of generality, in the present analysis we shall concentrate on the analysis of symmetric mode shapes. According to Eqs. (5) and (17), Fi and Fqi should be a periodic function of y with a period of 2p, i.e. 1 X

Fi ¼

Fim cosmy, Fqi ¼

m¼0

1 X

Fqim cosmy:

ð24Þ

m¼0

Substituting Eqs. (14), (23) and (24) into Eq. (1) gives @2 Fqim 2

@x

þ

q q q 1 @Fim @2 Fim m2 Fim þ  ¼ 0: 2 2 x @x @z x

ð25Þ

According to Eqs. (18)–(22), the general solution of Fqim for each liquid sub-domain can be expressed in terms of an infinite series by using the method of separation of variables (Wang et al., 2010). For annular cylindrical liquid sub-domain Oi (p ¼0,1,y,M, i¼2p þ1), one has 1   h   i X

m 1 1 1 m  Im lpmn x þ k1pmn Km lpmn x þA1im0 x þ x F1im ¼ A1imn cos lpmn zbp d2m d3i þ A1i00 d1m , ð26Þ n¼1

F2im ¼ d4i

1 X

   i h ih  2 2 2 2 2 2 A2imn elmn z 1 þ e2lmn ðbp zÞ N0m lmn a Jm lmn x J0m lmn a Nm lmn x ,

ð27Þ

n¼1

where Jm and Nm are the first and second kind Bessel functions of order m, Im and Km are the first and second kind modified Bessel functions of order m, J0m and N0m indicate the derivatives of Jm and Nm with respect to x, respectively. d1m, d2m, d3i, d4i, l1pmn , k1pmn and l2mn satisfy the following equations: ( ( 1, ðm ¼ 0Þ 0, ðm ¼ 0Þ d1m ¼ , d2m ¼ , ð28Þ 0, ðma0Þ 1, ðma0Þ (

d3i ¼

l1pmn

0,

ði ¼ 2M þ 1Þ , ðia2M þ 1Þ

1,

8 ð2n1Þp < 2ðb b Þ , pþ1 p ¼ : b npb , ð p þ 1 pÞ

(

d4i ¼

1, 0,

ði ¼ 2M þ 1Þ , ðia2M þ1Þ

ð29Þ

,

ð30Þ

ðp ¼ M Þ ðp ¼ 0,1,:::,M1Þ

  1 1 I0m lpmn þ k1pmn K0m lpmn ¼ 0,

ð31Þ

    2 2 2 2 N0m lmn a J0m lmn J0m lmn a N0m lmn ¼ 0,

ð32Þ

in which, I0m and K0m indicate the derivatives of In and Kn with respect to x, respectively. For cylindrical liquid sub-domain Oi (p¼ 0,1,y,M, i¼ 2pþ 2), one has 1    X 1 1 m 2 1 F1im ¼ A1imn cos lpmn zbp Im lpmn x þ A1im0 x dm d3i þ A1i00 dm , ð33Þ n¼1

F2im

¼

1 X

2

A2imn ezlmn =a

!

2 2 lmn 2lmn ðbp zÞ=a 1 þe x , Jm

n¼1

F3im ¼ d4i

1 X

ð34Þ

a

!

2 2 2 l A3imn ezlmn =a 1 þ d5i e2lmn ðbp þ 1 zÞ=a Jm mn x ,

a

n¼1

ð35Þ

2

in which, d3i , d4i , d5i and lmn satisfy the following equations: ( ( ( 0, ði ¼ 2M þ 2Þ 0, ði ¼ 2Þ 1, ði ¼ 2M þ 2Þ d3i ¼ ,d ¼ ,d ¼ , 1, ðia2M þ 2Þ 4i 1, ðia2Þ 5i 1, ðia2M þ2Þ  2 J0m lmn ¼ 0:

ð36Þ

ð37Þ

In Eqs. (26)–(27) and (33)–(35), Aqimn (q¼1,2,3, i¼1,y,2Mþ2, m¼0,1,2,y,n¼0,1,2,y) are the undetermined coefficients which can be determined by Eqs. (6) and (10).

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5. Eigenfrequency equations Substituting Eqs. (14) and (23) into Eq. (10), the artificial interface conditions between Oi and Oi0 can be written as follows: @Fi @Fi0 ¼ , on Gk : @gk @gk

Fi ¼ Fi0 ,

ð38Þ

Substituting Eq. (24) into the above equations yields Qi X

Fqim ¼

q¼1

Q i0 X

Fqi0 m ,

q¼1

Q i0 Qi X X @Fqi0 m @Fqim ¼ @gk @gk q¼1 q¼1

on Gk ,

ð39Þ

0

in which, the ordered triple (i,i ,k) satisfies Eq. (9). Introducing Eqs. (14) and (23) into Eq. (13), the free surface conditions can be rewritten as   @Fi  L2 Fi z ¼ b ¼ 0 on Gk : ð40Þ @z z ¼ b Substituting Eq. (24) into the above equation yields  Qi  X @Fqim   L2 F2im  ¼ 0 on Gk ,  @z  z¼b q¼1

ð41Þ

z¼b

where the ordered pair (i,k)A{(2Mþ 1,2M þ3),(2Mþ2,2Mþ2)}. The undetermined coefficients Aqimn (q¼ 1,2,3, i¼1,y,2Mþ 2, m¼0,1,2,y, n ¼0,1,2,y) can be determined by substituting Eqs. (26), (27) and (33)–(35) into Eqs. (39) and (41). On the cylindrical artificial interface Gk, truncating n in the series up to N þ 1 yields N X

  h   i 1 1 1 Im lpmn a þ k1pmn Km lpmn a A1imn cos lpmn zbp

n¼1

 þ A1im0 am þ am d2m d3i þ A1i00 d1m N    i h ih  X 2 2 2 2 2 2 þ d4i A2imn elmn z 1 þe2lmn ðbp zÞ N0m lmn a Jm lmn a J0m lmn a Nm lmn a n¼1

¼

   1 1 A1i0 mn cos lpmn zbp Im lpmn a þ A1i0 m0 am d2m d3i0 þA1i0 00 d1m

N X n¼1 N X

þ



 2 2 2 A2i0 mn ezlmn =a 1 þe2lmn ðbp zÞ=a Jm lmn

n¼1

þ d4i0

N X



 2 2 2 A3i0 mn ezlmn =a 1 þ d5i0 e2lmn ðbp zÞ=a Jm lmn ,

ð42Þ

n¼1 N X

  h   i 1 1 1 1 I0m lpmn a þ k1pmn K0m lpmn a A1imn lpmn cos lpmn zbp

n¼1

 þ A1im0 mam1 1a2m d2m d3i N    X 1 1 1 ¼ A1i0 mn lpmn cos lpmn zbp I0m lpmn a þ A1i0 m0 mam1 d2m d3i0 ,

ð43Þ

n¼1 0

in which, p¼0,1,y,M, i¼2pþ2, i ¼2p þ1, k¼p þ1. On the circular artificial interface Gk, truncating n in the series up to N þ1 yields N X

 1 m A1imn ð1Þn Im lpmn x þ A1im0 x d2m d3i þ A1i00 d1m

n¼1 N X

þ

!

2 2 2 l A2imn ebp þ 1 lmn =a 1 þ e2lmn ðbp bp þ 1 Þ=a Jm mn x

a

n¼1





þ 1 þ d5i d4i

N X

2

A3imn ebp lmn =a Jm

n¼1

¼

N X n¼1

A1i0 mn Im



l1p þ 1mn x



2 lmn

a

!

x

m þ A1i0 m0 x d2m d3i0

þ A1i0 00 d1m þ2

N X n¼1

2

A2i0 mn ebp þ 1 lmn =a Jm

!

2

lmn

a

x

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

þ d4i0

!

2 2 2 l A3i0 mn ebp þ 1 lmn =a 1 þ d5i0 e2lmn ðbp þ 2 bp þ 1 Þ=a Jm mn x ,

N X

N 1X

¼

2

bp þ 1 lmn

2

an¼1

A2mn lmn e

N d4i0 X

a

ð44Þ

a

n¼1

a

145

!

2 2 l 1e2lmn ðbp bp þ 1 Þ=a Jm mn x

a

2 2 A3i0 mn lmn ebp þ 1 lmn =a

!

2 2 lmn 2 l b  b = a ð Þ p þ 2 p þ 1 mn 1d5i0 e x , Jm

ð45Þ

a

n¼1 0

in which, p ¼0,1,y,M 1, i¼2p þ2, i ¼2p þ4, k¼2p þ3. On the free surface G2M þ 2, truncating n in the series up to N þ1 yields !

2 N N  1X 2 2 X 2 lmn 1 1 A1 l ð1Þn Im l x þ A2 l ebM þ 1 lmn =a 1e2lmn ðbM bM þ 1 Þ=a J x imn Mmn

n¼1



þ

Mmn

an¼1



N 1d5i d4i X

a

L2m

2

A3imn lmn elmn bM þ 1 =a Jm

lmn

n¼1

N X

a

a

!

2

2

m

imn mn

x

!

2 2 2 l A2imn ebM þ 1 lmn =a 1 þe2lmn ðbM bM þ 1 Þ=a Jm mn x L2m A1i00 d1m ¼ 0,

a

n¼1

ð46Þ

in which, i ¼2Mþ2. On the free surface G2M þ 3, truncating n in the series up to N þ1 yields N X

h   i 1 1 1 A1imn lMmn ð1Þn Im lMmn x þ k1Mmn Km lMmn x

n¼1

þ d4i

 h ih  2 2 2 2 2 A2imn lmn elmn bM þ 1 1e2lmn ðbM bM þ 1 Þ N0m lmn a Jm lmn x

N X n¼1



N  i  h ih  X 2 2 2 2 2 2 J0m lmn a Nm lmn x L2m d4i A2mn elmn bM þ 1 1 þ e2lmn ðbM bM þ 1 Þ N0m lmn a Jm lmn x

J0m



l2mn



a

n¼1



i L2m A1i00 d1m ¼ 0,,

2 Nm lmn x

ð47Þ

in which, i¼2M þ1. Lm denotes the nondimensional frequency parameter corresponding to the circumferential wave 1 1 number m. Multiplying Eqs. (42) and (43) by cosðlpmn ðzbp ÞÞ (when p rM 1, one has n ¼ 0,1,2,. . .,N and lpm0 ¼ 0; when p ¼M, one has n ¼ 0,1,2,. . .,N) and making integral from bp to bp þ 1, the space coordinate z can be eliminated. Multiplying 2 2 Eqs. (44)–(46) by J m ðlmn x=aÞ (when m¼0, one has n ¼ 0,1,2,:::,N and l00 ¼ 0; when ma0, one has n ¼ 0,1,2,:::,N) and making integral from 0 to a, the space coordinate x can be eliminated. Multiplying Eq. (47) by 2 2 2 2 2 ½N0m ðlmn aÞJm ðlmn xÞJ0m ðlmn aÞNm ðlmn xÞ (when m ¼0, one has n ¼ 0,1,2,. . .,N and l00 ¼ 0; when ma0, one has n ¼ 0,1,2,. . .,N) and making integral from a to 1, the space coordinate x can be eliminated. The system of linear equations for the undetermined coefficients {Am} can be written in the form of h i ð48Þ Dm L2m Dm fAm g ¼ 0,   in which, the matrixes [Dm], Dm and the vector {Am} are " 11 # " # 0 0 Dm D12   m 21 22 , ½Dm  ¼ , Dm ¼ Dm Dm D21 D22 m m fAm g ¼

ð49Þ

hn on o n on on o A11mn , A12mn ,. . ., A12M þ 2mn , A22M þ 1mn , A22mn , n on on on o n on oiT  A34mn , A24mn , A36mn , A26mn ,. . ., A32M þ 2mn , A22M þ 2mn ,,

where 2 6 6 6 6 D11 m ¼6 6 4

3

W1 W2

0 &

0

WM WM þ 1

7 7 7 7, 7 7 5

ð50Þ

146

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

2 D12 m

6 6 6 ¼6 6 6 4 2

6 6 6 21 Dm ¼ 6 6 6 4 2 6 6 6 6 ¼ D22 6 m 6 6 4

3

B01 0

B02

^

&

&

0



0

B0M

BM þ 1

0



0

WM þ 2

W0M þ 2

0

WM þ 3

B0M þ 1

7 7 7 7, 7 7 5 3

W0M þ 3

0

&

& W2M þ 1

0 BM þ 2

B0M þ 2

0

BM þ 3

B0M þ 3

0

^

&

&

&

0



0

B2M þ 1

B1S

0

 3 0 0 7 7 7 ^ 7 7, 0 7 5 WS 3 0 0 7 7 7 ^ 7: 7 0 7 5 2 BS

0

2

0 60 6 6 21 Dm ¼ 6 6^ 60 4 0 2 0 6 0 6 6 22 ^ Dm ¼ 6 6 6 0 4

1

BS

0



0

0



0

^

&

^

0



0

0



0

0



0

0



0

^

&

^

0



0

0



0

7 7 7 7, 7 0 W2M þ 1 7 5 WS 3

7 7 7 7 7, 7 0 B2M þ 1 7 5 2 BS

ð51Þ

  Every elements in matrixes [Dm] and Dm are the coefficients in Fourier series and Bessel series expansions, which can be numerically   calculated by the Gaussian integrals. The detailed expressions of {Am} and nonzero elements in matrixes [Dm] and Dm are given in Appendix A. For a given circumferential wave number m (m¼0,1,2,y), the nondimensional frequency parameter Lml can be determined by solving the linear eigenvalue eq. (48). It should be noted that the node number in the radial direction for the mode to the lth nondimensional frequency parameter Lml is just l.  corresponding  Moreover, although the matrixes [Dm] and Dm are not symmetric, no imaginary eigenvalue was found in our calculations. This is reasonable physically because the liquid damping was not considered. It should be noted that the numerical computations may be unstable for some limit cases such as 0 o a o0.05 (the extremely slender cylindrical liquid subdomain), 0.95o a o1 (the extremely narrow annular cylindrical liquid sub-domain) and 0 o bp þ 1  bp o0.05 (the extremely thin liquid layer). Substituting coefficients {Am} into Eqs. (26), (27) and (33)–(35) gives the solution of each sub-domain velocity potential. Introducing the sloshing height function f, one can rewrite the equation of the free surface waves in the form of fþ

1 @f ¼ 0 on Gf : g @t

ð52Þ

When the liquid is undergoing free sloshing in the container, f(r,y,t) is a harmonic function of time. According to Eqs. (24) and (52), f(r,y,t) should take the form of

f ðr, y,tÞ ¼

1 ejot X F m ðr Þcosmy, R2 m ¼ 0

ð53Þ

in which, Fm(r) is the radial surface profile of the liquid corresponding to the mode Fm. Substituting Eqs. (14), (23) and (53) into Eq. (52) gives F m ðxÞ ¼ L2m Fm ðx, bÞ, ðm ¼ 0,1,2,. . .Þ:

ð54Þ

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

147

6. Convergence and validation 6.1. Convergence study In order to demonstrate the computational stability and the convergent rate of the proposed method, a convergence study on nondimensional frequency parameters was carried out. According to Eqs. (27), (28) and (34)–(36), frequency parameters depend on the dimensionless parameters M, a, bp (p¼ 1,2,y,Mþ1). In the following computations, the ratio of liquid height and container radius was fixed at bM þ 1 ¼1 unless stated. The number of baffles was taken to be M¼2,3,4, respectively. For M¼ 2, two baffles were positioned at b1 ¼0.6 and b2 ¼0.8, respectively. The inner–outer radius ratio of the baffles was taken as a ¼0.3. For M¼3, three baffles were positioned at b1 ¼0.4, b2 ¼0.6 and b3 ¼0.8, respectively. The inner–outer radius ratio of the baffles was taken as a ¼0.5. For M¼4, four baffles were positioned at b1 ¼0.2, b2 ¼0.4, b3 ¼0.6 and b4 ¼0.8, respectively. The inner–outer radius ratio of the baffles was taken as a ¼0.7. Tables 1–3 give the convergence of the frequency parameters L2ml (m¼0,1,2, l ¼1,2,3,4) with respect to the number of terms for M¼2,3,4, respectively. It is seen from Tables 1–3 that the frequency parameters converge rapidly with the increase of the number of the terms. Three significant figures can be obtained when the number of terms truncated is more than 12. Four significant figures can be observed when the number of terms used is more than 20. Therefore, in all the following calculations, the number of the terms is fixed at 20.

6.2. Comparison study In order to check the validity of the present method, a finite element analysis was carried out by using the commercial software ADINA. The liquid in ADINA model was built by the 3-D potential-based liquid element with 27 nodes. Two types of potential-interfaces (free surface and rigid-wall) were used to simulate the boundary conditions of the liquid. Ne denotes the number of the potential-based liquid elements. The number of baffles was taken as M¼1,2,3, respectively. For M ¼1, the baffle was positioned at b1 ¼0.75 and the inner–outer radius ratio of the baffle was taken as a ¼0.7. Three different Table 1 Convergence of L2ml (m¼ 0,1, l ¼1,2,3,4) versus the number of terms N for parameters a ¼ 0.3, M ¼2 (b1 ¼ 0.6, b2 ¼0.8). m

l

N ¼4

N¼ 6

N¼8

0

1 2 3 4

2.491 6.312 9.926 13.24

2.491 6.317 9.932 13.24

2.491 6.318 9.933 13.24

1

1 2 3 4

0.7072 4.501 8.210 11.69

0.7076 4.504 8.203 11.59

2

1 2 3 4

1.677 5.948 9.708 13.14

1.677 5.946 9.709 13.10

N ¼10

N ¼ 12

N ¼ 16

N ¼20

N ¼24

2.491 6.318 9.933 13.24

2.492 6.318 9.933 13.24

2.492 6.318 9.933 13.24

2.492 6.318 9.932 13.24

2.492 6.318 9.932 13.24

0.7080 4.506 8.201 11.58

0.7083 4.507 8.200 11.57

0.7085 4.508 8.199 11.56

0.7088 4.509 8.199 11.56

0.7089 4.510 8.199 11.56

0.7089 4.510 8.199 11.56

1.677 5.946 9.710 13.09

1.677 5.946 9.710 13.09

1.677 5.946 9.710 13.08

1.677 5.947 9.711 13.08

1.677 5.947 9.711 13.08

1.677 5.947 9.711 13.08

Table 2 Convergence of L2ml (m¼ 0,1, l ¼1,2,3,4) versus the number of terms N for parameters a ¼ 0.5, M ¼3 (b1 ¼ 0.4, b2 ¼ 0.6, b3 ¼0.8). m

l

N ¼4

N¼6

N¼8

N¼ 10

N ¼12

N ¼ 16

N ¼ 20

N ¼ 24

0

1 2 3 4

2.641 6.638 9.966 13.25

2.649 6.644 9.986 13.26

2.651 6.647 9.990 13.26

2.653 6.648 9.992 13.26

2.654 6.649 9.992 13.26

2.655 6.650 9.993 13.26

2.656 6.651 9.993 13.26

2.656 6.651 9.993 13.26

1

1 2 3 4

0.8819 4.855 8.259 11.61

0.8853 4.855 8.255 11.61

0.8871 4.855 8.255 11.61

0.8883 4.856 8.255 11.61

0.8891 4.856 8.254 11.61

0.8910 4.856 8.254 11.61

0.8911 4.857 8.254 11.61

0.8911 4.857 8.254 11.61

2

1 2 3 4

1.787 6.301 9.788 13.11

1.789 6.305 9.786 13.11

1.791 6.307 9.786 13.11

1.792 6.308 9.785 13.11

1.792 6.309 9.784 13.11

1.793 6.310 9.784 13.11

1.794 6.311 9.783 13.11

1.794 6.311 9.783 13.11

148

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

Table 3 Convergence of L2ml (m¼ 0,1, l ¼ 1,2,3,4) versus the number of terms N for parameters a ¼ 0.7, M ¼ 4 (b1 ¼ 0.2, b2 ¼0.4, b3 ¼ 0.6, b4 ¼ 0.8). m

l

N ¼4

N¼6

N ¼8

N ¼10

N ¼12

N ¼ 16

N¼ 20

N ¼ 24

0

1 2 3 4

3.082 6.760 10.08 13.27

3.095 6.764 10.09 13.28

3.101 6.764 10.09 13.29

3.105 6.764 10.09 13.29

3.107 6.764 10.09 13.29

3.110 6.764 10.09 13.29

3.112 6.764 10.09 13.29

3.112 6.764 10.09 13.29

1

1 2 3 4

1.225 4.939 8.408 11.65

1.232 4.938 8.405 11.65

1.236 4.938 8.404 11.65

1.238 4.938 8.403 11.65

1.240 4.938 8.403 11.65

1.242 4.938 8.403 11.65

1.243 4.938 8.402 11.65

1.243 4.938 8.402 11.65

2

1 2 3 4

2.161 6.469 9.881 13.13

2.170 6.470 9.882 13.13

2.175 6.470 9.882 13.13

2.178 6.469 9.882 13.13

2.180 6.469 9.882 13.13

2.183 6.468 9.883 13.13

2.185 6.468 9.883 13.13

2.185 6.468 9.883 13.13

Table 4 Comparison study of L2ml for parameters: M¼ 1 (b1 ¼0.5, a ¼ 0.7), M¼ 2 (b1 ¼0.5, b2 ¼ 0.75, a ¼ 0.5), M ¼3 (b1 ¼ 0.25, b2 ¼0.5, b3 ¼ 0.75, a ¼ 0.5). M

1

Method

ADINA

Ne

1, 1

2, 1

0, 1

3, 1

4, 1

1, 2

5, 1

2, 2

0, 2

6, 1

1992 4940 9484

1.628 1.625 1.622 1.624 1.621

2.909 2.907 2.905 2.906 2.904

3.765 3.760 3.757 3.759 3.756

4.121 4.119 4.118 — 4.117

5.281 5.279 5.278 — 5.276

5.318 5.312 5.308 5.312 5.306

6.406 6.400 6.397 — 6.396

6.709 6.702 6.696 6.701 6.693

7.021 7.015 7.008 7.011 7.004

7.508 7.497 7.494 — 7.492

1648 4800 8800

1.034 1.031 1.029 1.026

2.086 2.084 2.080 2.076

3.008 3.002 2.997 2.994

3.344 3.333 3.329 3.327

4.670 4.663 4.660 4.636

5.064 5.060 5.053 5.049

5.985 5.979 5.973 5.921

6.525 6.521 6.518 6.503

6.883 6.857 6.848 6.832

7.262 7.252 7.241 7.158

1765 4512 8500

1.033 1.031 1.028 1.026

2.085 2.081 2.079 2.076

3.007 3.002 2.997 2.994

3.345 3.335 3.330 3.327

4.670 4.665 4.658 4.636

5.064 5.059 5.055 5.048

5.985 5.981 5.976 5.921

6.524 6.519 6.513 6.503

6.871 6.862 6.857 6.832

7.262 7.250 7.243 7.158

Gavrilyuk et al. (2006) Present 2

ADINA

Present 3

ADINA

Present

Circumferential wave number, radial wave number

discretizations were considered: Ne ¼1531, 4608, 8180, respectively. For M¼ 2, two baffles were positioned at b1 ¼0.5 and b2 ¼0.75, respectively. The inner–outer radius ratio of the baffles was taken as a ¼0.5. Three different discretizations were considered: Ne ¼1765, 4512, 8500, respectively. For M¼3, three baffles were positioned at b1 ¼0.25, b2 ¼0.5 and b3 ¼0.75, respectively. The inner–outer radius ratio of the baffles was taken as a ¼0.5. Three different discretizations were considered: Ne ¼ 1648, 4800, 8800, respectively. In Table 4, the first ten nondimensional frequency parameters obtained by the present method were compared with those obtained by ADINA for different numbers of elements. The results obtained by Gavrilyuk et al. (2006) for M ¼1 were also given for comparison. It is seen from Table 4 that the present results are in good agreement with those from ADINA and Gavrilyuk et al. (2006).

7. Numerical results 7.1. Natural frequencies versus the position of the baffle In this section, the effect of the position of the baffle on the nondimensional frequency parameters L2ml (m¼0,1, l ¼1) is investigated for two cases. In the first case, the number of the baffles is taken as M¼1,2, respectively. For M ¼2, the lower baffle is positioned at b1 ¼0.4. Two different inner–outer radius ratios of the baffles were taken as a ¼0.6,0.8, respectively. Figs. 3 and 4 depict the frequency parameters L211 and L201 with respect to the position of the upper baffle in the range of 0.45r b1 r0.9 for a ¼0.6 and a ¼0.8, respectively. It is seen from Figs. 3 and 4 that the frequency parameters monotonically decrease with respect to the upper baffle moving towards the free surface. Moreover, it is observed from Figs. 3 and 4 that the effect of the lower baffle increases with the inner radius of the baffles although the effect of the lower baffle on frequency parameters is not considerable. In the second case, the number of baffles is taken as M¼2 and two different inner–outer radius ratios of the baffles are taken as a ¼0.7,0.9. The upper baffle is positioned at b2 ¼ 0.8 and the lower baffle moves from b1 ¼0.1 to b1 ¼ 0.75.

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

Fig. 3. Nondimensional frequency parameters L2ml (m¼0,1, l ¼ 1) for M¼ 1 (a ¼0.6) and M¼ 2 (a ¼0.6, b1 ¼0.4).

Fig. 4. Nondimensional frequency parameters L2ml (m¼ 0,1, l ¼1) for M ¼1 (a ¼0.8) and M ¼2 (a ¼0.8, b1 ¼ 0.4).

Fig. 5. Nondimensional frequency parameters L2ml (m¼ 0,1, l ¼1) for M ¼2 (a ¼ 0.7, b2 ¼ 0.8).

149

150

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

Fig. 6. Nondimensional frequency parameters L2ml (m ¼0,1, l ¼ 1) for M ¼ 2 (a ¼ 0.9, b2 ¼0.8).

Fig. 7. Nondimensional frequency parameters L2ml (m ¼0,1, l ¼ 1) for two baffles versus the inner–outer radius ratio of the baffles for b1 ¼ 0.4,0.8 and b2 ¼0.9.

Figs. 5 and 6 depict the nondimensional frequency parameters with respect to the position of the lower baffle for a ¼0.7 and a ¼0.9, respectively. It is seen from Figs. 5 and 6 that the frequency parameter decreases as the lower baffle moves up from the bottom of the container. However, when the lower baffle further moves up to be near to the upper baffle the frequency parameter turns to increasing. For example, the frequency parameter L211 decreases with the increase of b1 up to b1 ¼0.67 then turns to increasing with the further increase of b1. The frequency parameter L201 decreases with the increase of b1 up to b1 ¼0.71 then turns to increasing with the further increase of b1. This means when the lower baffle is near to the upper baffle the effect of the lower baffle becomes very small. In such a case, the results for two baffles (M ¼2) close to those for single baffle (M¼ 1). 7.2. Natural frequencies versus the inner radius of the baffle The effect of the inner radius of the baffles on the nondimensional frequency parameters L2ml (m¼0,1, l ¼1) is investigated for two cases. In the first case, the upper baffle is positioned at b2 ¼0.9. Two different positions of the lower baffle are considered: b1 ¼0.4 and 0.8, respectively. The frequency parameters with respect to the inner–outer radius ratio of the baffles are plotted in Fig. 7. It is seen from Fig. 7 that the frequency parameters monotonically increase with the inner radius of the baffle. This means the effect of the baffles decreases with the increase of the inner radius of the baffles. Moreover, the difference of the frequency parameters between b1 ¼0.4 and 0.8 is slight. This implies that the effect of the position of the lower baffle is very small.

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

151

Fig. 8. Nondimensional frequency parameters L2ml (m¼0,1, l ¼ 1) for two baffles versus the inner–outer radius ratio of the baffles for b1 ¼ 0.5 and b2 ¼ 0.7,0.9.

Fig. 9. Radial surface profiles Fml (m¼ 0,1, l ¼ 1) of the sloshing modes in the rigid cylindrical container with two baffles for a ¼ 0.65, b1 ¼ 0.6 and b2 ¼ 0.7,0.8,0.9.

In the second case, the lower baffle is positioned at b1 ¼0.5. Two different positions of the upper baffle are considered: b2 ¼0.7 and 0.9, respectively. The nondimensional frequency parameters with respect to the inner–outer radius ratio of the baffles are depicted in Fig. 8. As shown in Fig. 8, the frequency parameters monotonically increase with the inner radius of the baffle. The difference of the frequency parameters between b2 ¼0.7 and 0.9 is significant for small inner–outer radius ratio. However, the difference decreases with the increase of the inner radius of the baffles. This implies that the effect of the position of the upper baffle decreases with increase of the inner–outer radius ratio of the baffles. Especially, when the inner–outer radius ratio of the baffles closes to 1 the effect of the baffles vanishes. 7.3. Radial surface profiles of natural modes Sloshing mode shapes corresponding to L2ml (m¼0,1, l ¼1) have been investigated according to Eq. (54). The radial surface profiles responding to L2ml are described by Fml(x). The number of the baffles is taken as M ¼2. The radial surface profiles Fml(x) (m ¼0,1, l¼1) for parameters a ¼0.65, b1 ¼0.6 and b2 ¼0.7,0.8,0.9 are shown in Fig. 9. It is seen from Fig. 9 that the position of the upper baffle has a considerable influence on the mode shapes. In Fig. 10, the radial surface profiles Fml(x) (m¼ 0,1, l ¼1) are plotted for parameters a ¼0.7, b1 ¼0.3,0.7 and b2 ¼0.9. The radial surface profiles given by

152

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

Fig. 10. Radial surface profiles Fml (m ¼0,1, l ¼ 1) of the sloshing modes for M¼ 2 (a ¼0.7, b1 ¼0.3,0.7, b2 ¼ 0.9).

Fig. 11. Radial surface profiles Fml (m¼ 0,1, l ¼ 1) of the sloshing modes in the rigid circular cylindrical container with two baffles for a ¼0.4,0.6,0.8, b1 ¼0.7 and b2 ¼0.9.

Gavrilyuk et al., (2006) for M¼1 (a ¼0.7, b1 ¼ 0.9) are also depicted for comparison. It is seen from Fig. 10 that the effect of the position of the lower baffle on the mode shapes is very small and can be ignored. The radial surface profiles Fml(x) (m¼0,1, l ¼1) for three different inner–outer radius ratios of the baffle: a ¼ 0.4,0.6,0.8 are indicated in Fig. 11 when two baffles are positioned at b1 ¼0.7 and b2 ¼ 0.9, respectively. It is seen from Fig. 11 that the inner radius of the baffles has considerable effect on the mode shapes. 8. Conclusions In this paper, a semi-analytical approach has been developed to obtain the natural frequencies and mode shapes of liquid sloshing in a rigid cylindrical container with multiple baffles of the same inner radius. The liquid domain is divided into several simple liquid sub-domains so that liquid velocity potential in each sub-domain is of class C1 with continuity boundary conditions. Based on the superposition principle, the analytical solutions of the liquid velocity potential in each liquid sub-domain are obtained by using the method of separation of variables. The eigenfrequency equation is derived from the free surface wave condition and the continuity interface conditions by using the Fourier series expansion in liquid height direction and Bessel series expansion in radial direction. The convergence and comparison studies showed that the

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

153

solution accurate up to four significant figures could be obtained by using the series of 20 terms. The natural frequencies and mode shapes versus the position, the inner–outer radius ratio and the number of annular baffles are discussed in detail. The following observations are highlighted: 1. The nondimensional frequency parameters monotonically decrease with respect to the uppermost baffle moving towards the free surface. 2. For multiple baffles, the variation of the nondimensional frequency parameters is non-monotonic with respect to the position of the lower baffle. 3. The natural frequencies increase with the increase of the inner radius of the baffles. The effects of the uppermost baffle on natural frequencies and modes are much more significant than the other baffles.

Acknowledgments The financial supports from National Natural Science Foundation of China, Grant no. 11172123 and from the Research Grant Council of the Hong Kong Special Administration Region, China, for the research project 10500050 are greatly acknowledged. Appendix A The detailed expressions of {Am} are  q   q  Aimn ¼ Aim1 ,Aqim2 ,Aqim3 ,:::,AqimN , ðq ¼ 1,2,3,i ¼ 1,. . .,2M þ 2Þ:

ðA1Þ

0

On the cylindrical interface Gk (k¼pþ 1, i¼2p þ2, i ¼2p þ1, p¼ 0,1,y,M), the corresponding nonzero elements in [Dm(Lm)] are given as follows: 2 3 "  11   12  # ½ 0 ½0 wkn0 n wkn0 n h i h i 0 5, ðA2Þ Wk ¼  21   22  , Bk ¼ 4 021 022 bkn0 n bkn0 n wkn0 n wkn0 n 2 Bk ¼

4h

½0 21

bkn0 n

i

½0 ½0

3 5, ðk ¼ M þ1Þ,

ðA3Þ

where n h   i 8

o 1 1 0 0 > < d~ 3n0 n d~ 1mk Im lpmn0 a þ k1pmn0 Km lpmn0 a þ d~ 2mk am1 1a2m ,n0 ¼ 1 h   i w11 , kn0 n ¼ > ~ 1 1 : d 3n0 n I0m lpmn0 d~ a þ k1pmn0 K0m lpmn0 d~ a ,n0 4 1 2mk

2mk

ðn ¼ 1,2,. . .,N þ 1d~ k , n ¼ 1,2,. . .,N þ d~ 2mk Þ, 0

ðA4Þ

h  i 8 1 0 > < d~ 3n0 n d~ 1mk Im lpmn0 a þ d~ 2mk am1 ,n0 ¼ 1  w12 , kn0 n ¼ > ~ : d 3n0 n I0m l1pmn0 d~ a ,n0 41 2mk ðn ¼ 1,2,. . .,N þ 1d~ k , n0 ¼ 1,2,. . .,N þ d~ 2mk Þ,

ðA5Þ

  8   R bp þ 1 1 > < d~ 4n0 n d1m þ ðam þ am Þd2m d3i bp cos lpmn0 1 þ d~ 1k zbp dz, ðn0 ¼ 1Þ h   i R b   , w21 kn0 n ¼ > ~ 1 pþ1 : d 4n0 n Im l1pmn0 a þ k1imn0 Km l1pmn0 a cos2 lpmn0 1 þ d~ zbp dz, ðn0 41Þ b p

1k

ðn ¼ 1,2,. . .,N þ 1d~ 1k , n0 ¼ 1,2,. . .,N þ 1d~ 1k Þ,

ðA6Þ

  8 R bp þ 1 1 > < d~ 4n0 n ðd1m þ am d2m d3i Þ bp cos lpmn0 1 þ d~ 1k zbp dz, ðn0 ¼ 1Þ  Rb   , w22 kn0 n ¼ > ~ : d 4n0 n Im l1pmn0 a b p þ 1 cos2 l1pmn0 1 þ d~ zbp dz, ðn0 4 1Þ 1k p

ðn ¼ 1,2,. . .,N þ 1d~ 1k , n0 ¼ 1,2,. . .,N þ 1d~ 1k Þ,

ðA7Þ



 2 Z bp þ 1   2 2 021 1 ezlmn =a 1þ d5i0 e2lmn ðbp þ 1 zÞ=a cos lpmn0 1 þ d~ 1k zbp dz, bkn0 n ¼ d4i0 Jm lmn bp

ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,N þ1d~ 1k Þ,

ðA8Þ

154

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

 2 Z 022 bkn0 n ¼ J m lmn

bp þ 1

bp



  2 2 1 ezlmn =a 1 þe2lmn ðbp zÞ=a cos lpmn0 1 þ d~ 1k zbp dz,

ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,N þ 1d~ 1k Þ,

d4i

Z

zti zbi

ðA9Þ



b    2 2 1 b elmn z 1 þe2lmn zi z cos lpmn0 1 þ d~ 1k zzi dz ðn ¼ 1,2,. . .,N,n0 ¼ 1,2,. . .,N þ 1d~ 1k Þ,

in which, d~ 1mk , d~ 2mk , d~ 3nn0 ,d~ 4nn0 and d~ 1k satisfy 8 > < 1, ðm ¼ 0,k ¼ 1,2,. . .,M1Þ ~d 0, ðma0,k ¼ 1,2,. . .,M1Þ , ¼ 1mk > : 1, ðk ¼ MÞ 8 0, ðm ¼ 0,k ¼ 1,2,. . .,M1Þ > < ~d 1, ðma0,k ¼ 1,2,. . .,M1Þ , 2mk ¼ > : 0, ðk ¼ MÞ ( ( 1, ðn0 ¼ nÞ 1, k ¼ M ~d 0 ¼ ~ , d 1k ¼ , 4n n 0, ðn0 anÞ 0, kaM

ðA10Þ

(

d~

3n0 n

¼

ðn0 ¼ n1þ d~ 2mk þ d~ 1k Þ : þ d~ Þ ðn0 an1 þ d~

1, 0,

2mk

ðA11Þ

1k

0

On the circular interface Gk (k¼p þMþ2, i¼2p þ2, i ¼2p þ1, p ¼0,1,y,M  1), the corresponding nonzero elements in [Dm(Lm)] are given as follows: " # " # ½0 ½0 ½ 0 ½0 0     , Wk ¼ , Wk ¼ ½0 w22 ½0 w022 kn0 n kn0 n h i3 i 2 2h 3 12 011 ½0 ½0 bkn0 n bkn0 n 6 7 6 7 ðA12Þ Bk ¼ 4 h 21 i h 22 i 5, B0k ¼ 4 h 021 i h 022 i 5, bkn0 n bkn0 n bkn0 n bkn0 n where

w22 kn0 n ¼

8 R  > > > a x d1m þ xm d2m d3i Jm > 0 > <  > Ra > 1 > > ð1Þn 0 xIm lpmn x Jm > :

!

2

lmn0 d

a

1m

2 lmn0 d 1m

a

x dx, ðn ¼ 1Þ !

,

x dx, ðn 4 1Þ

ðn ¼ 1,2,. . .,N þ 1, n0 ¼ 1,2,. . .,N þ d1m Þ,

ðA13Þ

! 8 2  i R h > l > > a x d2m d~ 2k Im l1p þ 1mn x þ d1m þ xm d2m d3i Jm mn0 d1m x dx, ðn ¼ 1Þ > a > < 0 ! , w022 kn0 n ¼ 2  > Ra l 0 > > > ð1Þn 0 xIm l1p þ 1mn x Jm mn ad1m x dx, ðn 4 1Þ > : ðn ¼ 1,2,. . .,N þ 1d~ 2k , n0 ¼ 1,2,. . .,N þ d1m Þ,

ðA14Þ !



Z a 2 2 2 l 0 12 xJ2m mn x dx bkn0 n ¼ d~ 5n0 n ebp þ 1 lmn0 =a 1e2lmn0 ðbp bp þ 1 Þ=a

a

0

ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,NÞ,

ðA15Þ

!

Z a 2 2 2 l 0 011 xJ2m mn x dx, bkn0 n ¼ d~ 5n0 n d4i0 ebp þ 1 lmn0 =a 1d5i0 elmn0 ð2bp þ 2 bp þ 1 Þ=a 0

a

ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,NÞ,

ðA16Þ

! 2 Z a  2 l 21 xJ2m mn x dx  ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,N þ d1m Þ, bkn0 n ¼ d~ 6n0 n d4i 1þ d5i ebp lmn =a 0

22 bkn0 n

¼ d~

2

bp þ 1 lmn =a

6n0 n e

a

ðA17Þ

!

Z a 2 2 lmn 2 2lmn ðbp bp þ 1 Þ=a 1þ e xJm x dx 0

0

ðn ¼ 1,2,. . .,N, n ¼ 1,2,. . .,N þ d1m Þ,

a

ðA18Þ

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

!

Z a 2 2 2 l 021 bkn0 n ¼ d~ 6n0 n d4i0 ebp þ 1 lmn =a 1 þ d5i0 e2lmn ðbp þ 2 bp þ 1 Þ=a xJ2m mn x dx,  ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,N þ d1m Þ,

a

0

Z a

2

bkn0 n ¼ 2d~ 6n0 n ebp þ 1 lmn =a 022

0

a

ðA19Þ

!

2

lmn

xJ2m

155

x dx,  ðn ¼ 1,2,. . .,N, n0 ¼ 1,2,. . .,N þ d1m Þ,

in which, d~ 2k , d~ 5n0 n and d~ 6n0 n satisfy ( ( 1, 1, ðk ¼ M1Þ ~d ¼ ~ ,d 0 ¼ 2k 0, 0, ðkaM1Þ 5n n

ðn ¼ n0 Þ 0

ðnan Þ

( , d~

6n0 n

¼

1,

ðn ¼ n0 þ1d1m Þ

0,

ðnan0 þ 1d1m Þ

:

On the free surface Gf, the corresponding nonzero elements in [Dm(Lm)] are given as follows: i i 2h 3 2h 3 ½0 ½0 w11 0 w11 k 1 n0 n 6 k1 n n 7 6 h i 5,WS ¼ 4 h i7 WS ¼ 4 5, 22 ½ 0 ½  w22 0 w 0 0 k2 n n k2 n n 2 h 11 i 3 2h 3 i 11 ½0 bk1 n0 n ½0 bk1 n0 n 1 1 4 5, 4 5 ,BS ¼ BS ¼ ½0 ½0 ½0 ½0 2 3 2 3 ½0 ½ 0 ½0 ½0 h i h i5 2 4 h 22 i 5 2 4 , ðk1 ¼ 2M þ 3,k2 ¼ 2M þ 2Þ, ,BS ¼ BS ¼ 21 22 bk2 n0 n bk2 n0 n ½0 bk2 n0 n

ðA20Þ

ðA21Þ

ðA22Þ

where Z 1 h     i n 1 x N0m l2mn0 d1m a Jm l2mn0 d1m x J0m l2mn0 d1m a Nm l2mn0 d1m x w11 k1 n0 n ¼ ð1Þ lMmn a n h   i o 1 1  d2m Im lMmnd1m x þ k1Mmn Km lMmnd1m x þ d1m dx, ðn ¼ 1,2,. . .,N þ d1m , n0 ¼ 1,2,. . .,N þ d1m Þ, ( w11 k1 11 ¼

ð1a2 Þd1m 2

0

(

ðA23Þ

a2 d1m

,w22 k2 11 ¼

2

0

,

ðA24Þ

22 0 w11 k1 n0 n ¼ wk2 n0 n ¼ 0, ðn ¼ 2,. . .,N þ d1m , n ¼ 2,. . .,N þ d1m Þ,

ðA25Þ

h i 2 2 11 2 bk1 n0 n ¼ d~ 7n0 n d4i lmn elmn bp þ 1 1e2lmn ðbp bp þ 1 Þ Z 1     i2 x N0m l2mnd1m a Jm l2mnd1m x J0m l2mnd1m a Nm l2mnd1m x dx  a

ðn ¼ 1,2,. . .,N þ d1m , n0 ¼ 1,2,. . .,N þ d1m Þ,

ðA26Þ

h i 2 2 11 bk1 n0 n ¼ d~ 7n0 n d4i elmn bp þ 1 1 þ e2lmn ðbp bp þ 1 Þ Z 1     i2 x N0m l2mnd1m a Jm l2mnd1m x J0m l2mnd1m a Nm l2mnd1m x dx,  a

ðn ¼ 1,2,. . .,N þ d1m , n0 ¼ 1,2,. . .,N þ d1m Þ, w22 k2 n0 n

1 ¼ ð1Þn lMmn

Z a 0

0

xJm @

1

2

lmn0 d1m

a



ðA27Þ







xA d2m Im l1Mmnd1m x þ d1m dx,

0

ðn ¼ 1,2,. . .,N þ d1m , n ¼ 1,2,. . .,N þ d1m Þ,

21 bk2 n0 n

¼

 2 1d5i d~ 7n0 n d4i lmn

a

2

lmn bp þ 1 =a

e

Z a 0

2

0

x4Jm @

ðA28Þ 2

lmnd1m

a

132

xA5 dx,

ðn ¼ 1,2,. . .,N þ d1m , n0 ¼ 1,2,. . .,N þ d1m Þ, 2

22 bk2 n0 n

¼

d~ 7n0 n lmn

a

2

bp þ 1 lmn =a

e

ðA29Þ

2 0 2 132

Z a 2 lmnd1m 2lmn ðbp bp þ 1 Þ=a 1e x4Jm @ xA5 dx, 0

0

ðn ¼ 1,2,. . .,N þ d1m , n ¼ 1,2,. . .,N þ d1m Þ,

a

ðA30Þ

156

J.D. Wang et al. / Journal of Fluids and Structures 34 (2012) 138–156

22 bk2 nn

¼ d~

2

bp þ 1 lmn =a

7n0 n e



2

2lmn ðbp bp þ 1 Þ=a

1þ e

Z a 0

2

0

x4Jm @

ðn ¼ 1,2,. . .,N þ d1m , n0 ¼ 1,2,. . .,N þ d1m Þ:

2

lmnd1m

a

132

xA5 dx, ðA31Þ

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