Research Article  Open Access
Nonlinear Sloshing of Liquid in a Rigid Cylindrical Container with a Rigid Annular Baffle under Lateral Excitation
Abstract
Nonlinear response of liquid partially filled in a rigid cylindrical container with a rigid annular baffle subjected to lateral excitation is studied. A semianalytical approach is presented to determine the natural frequencies and modes of the liquid sloshing. Introducing the generalized timedependent coordinates, the surface wave height and the velocity potential are expressed in terms of the natural modes of liquid sloshing. Based on the Bateman–Luke variational principle, the infinitedimensional modal system is given by the variational procedure. The infinitedimensional modal system is reduced by using the Moiseev asymptotic relations. The resultant hydrodynamic force and moment of the liquid pressure acting on the container mainly depend on the position vector of the mass center of the liquid. Expanding the integral about the weighted position coordinates into the Taylor series about the surface wave height at the unperturbed free surface gives the formula of the position vector of the mass center, which depends only on the generalized timedependent coordinates. Excellent agreements have been achieved by comparing the present results with those obtained from Gavrilyuk’s solution and SPH solution. Finally, the surface wave height, resultant hydrodynamic force, and hydrodynamic moment for a container subjected to harmonic lateral excitation are discussed in detail.
1. Introduction
Liquid sloshing can cause a serious problem in liquid containers subjected to lateral excitation. For instance, the failure of the floating roof and the fire of the oil storage containers due to liquid sloshing were observed in some earthquake disasters [1]. As is well known, the baffle mounted in a partially liquidfilled container could provide the energy dissipation of the liquid [2], which reduces the sloshing amplitude. As a fringe benefit, the baffles could enhance the stiffness of the container.
Subjected to excitations, the linear dynamic response of liquid in a container equipped with baffles was investigated. Abramson [3] conducted some experiments to study the response of the liquid in a baffled fuel container, which is subjected to pitching and lateral excitations, respectively. Using the boundary element method, Gedikli and Ergüven [4] simulated the seismic response of the rigid liquidfilled container with an annular baffle. Isaacson and Premasiri [5] studied the dynamic damping due to baffles in a liquidfilled rectangular container or reservoir undergoing horizontal excitation. Using the finite element method, Biswal et al. [6] studied the effect of an annular baffle on the dynamic response of a partially liquidfilled cylindrical container. Based on the velocity potential formulation and linearization theory, Goudarzi et al. [7] developed an analytical model to estimate the dynamic damping ratio of liquid sloshing due to baffles in a liquidfilled rectangular container subjected to lateral excitation. Wang et al. [8, 9] developed a semianalytical method to solve the natural frequencies and modes of the liquid sloshing in the cylindrical container with single or multiple baffles. Based on the natural frequencies and modes obtained, the linear dynamic response of the liquid in a baffled container subjected to lateral excitation was analyzed [10, 11].
Generally, the nonlinear sloshing problems of liquid are described by the Navier–Stokes equations and the dynamic and kinematic conditions on the free surface [12]. The problems are usually investigated by the experimental and numerical methods [13–15]. Carra et al. [16] investigated the linear and geometrically nonlinear dynamical response of a thin plate in contact with liquid. They proposed the experimental method to study the behavior of the liquid’s free surface during the nonlinear vibration of the thin plate undergoing harmonic excitations. Wu et al. [17] developed a timeindependent finite difference scheme with fictitious cell technique to investigate viscous liquid sloshing in twodimensional containers with baffles. However, this description creates some mathematical and computational difficulties. Due to these difficulties, a lot of researchers are interested in using the potential principle as described by Zhou et al. [18].
Based on the potential formulations, there are two different approaches to study the nonlinear problems of liquid sloshing. The first is the numerical method [19, 20]. Hugo et al. [21] proposed the numerical method to investigate the sloshing response of cylindrical containers subjected to earthquake ground motion. Cho and Lee [22] and Cho et al. [23] simulated the large amplitude liquid sloshing in a twodimensional baffled rectangular container under lateral excitation. Using the finite element method, Biswal et al. [24] studied the twodimensional nonlinear sloshing in both rectangular and cylindrical containers equipped with rigid baffles.
The second is the multimodal method, namely, the free surface and velocity potential are expressed in terms of the generalized Fourier expansion. Substituting the expansions into the original freeboundary value equations yields the modal system. The linear and nonlinear multimodal methods, which can be used to conduct the analytical investigations of resonant sloshing regimes, secondary resonance, and chaos, are presented in [25]. It also includes the nonlinear multimodal method developed by Faltinsen et al. [26]. Using the Bateman–Luke variational statement, they derived the infinitedimensional modal system for the nonlinear sloshing of liquid in a container. Based on the Narimanov–Moiseev asymptotic relations, the infinitedimensional modal system can be reduced to the finitedimensional modal system. The method is applicable to arbitrary excitations and a broad class of the container shapes [27–30]. Lukovsky [31] developed the fivedimensional modal system for the nonlinear sloshing in the upright circular cylindrical container, which was validated by the experiments. Using the asymptotic modal method, Gavrilyuk et al. [32, 33] investigated the nonlinear resonant sloshing of liquid in a rigid circular cylindrical container with a rigid annular baffle. Raynovskyy and Timokha [34] studied the damped steadystate resonant sloshing in a circular base container by the nonlinear multimodal system, which is equipped with linear damping terms associated with the logarithmic decrements of the natural sloshing modes. It should be mentioned that the research on the multimodal system of containers equipped with baffles is insufficient.
Using the variational multimodal method, Zhou et al. [18] investigated the nonlinearity of the free sloshing of the liquid in a baffled container caused by some initial displacement of the free surface from the viewpoint of no external excitation. This calculation focused on the surface wave height as an indicator of the liquid sloshing. However, the hydrodynamic forces due to the liquid sloshing were not at all investigated. This can be a problem because if the excitation frequency is close to the natural frequency of the liquid, the resonantamplified impact force may cause damage to the container. Thus, the proposed study focuses on the nonlinear response of the forced sloshing of the liquid in a baffled container subjected to lateral harmonic or seismic excitation. By applying Taylor series expansion to the integral of the weighted position coordinates of the surface wave height, we derive formulae for the resultant hydrodynamic force and the moment of liquid pressure acting on the baffled container. Parametric analysis is applied to the resultant hydrodynamic force and moment, and the distribution of liquid pressure on the container wall is discussed. We also conduct steadystate analysis in the frequency domain.
2. Multimodal Method
A rigid cylindrical container with a thin rigid annular baffle, which is partially filled with inviscid, incompressible, and irrotational liquid, is depicted in Figure 1, where is the absolute coordinate system and is the relative cylindrical coordinate system fixed at the container. The origin of is located at the center of the bottom of the container, and the zaxis is orthogonal to the bottom of the container. The inner radius of the container is r_{2}, and the inner radius of the baffle is r_{1}. and denote, respectively, the positions of the baffle and the unperturbed (hydrostatic) free surface. The thickness of the baffle is very small compared with the inner radius of the container. The density of the liquid is . The timedependent liquid domain is bounded by the free surface , the wet container wall , the container bottom , and the baffle surface . The potential formulation of the nonlinear sloshing problem is given as follows [25]:where is the outer normal of the boundary of the liquid domain , is the timedependent shape equation of the free surface , is the translation velocity vector of the origin related to the coordinate system , is the angular velocity vector of the container, is the position vector with respect to , is the position vector of the point with respect to , is the position vector with respect to , and is the gravity acceleration vector, and equation (6) denotes the initial conditions. In the cylindrical coordinate system , the unit vectors codirectional with the longitudinal axis () and the zaxis () are and , respectively. The unit vector is given by . Then, the vectors in equations (1)–(6) can be written as
When no overturning wave occurs, the continuum problem is reduced to a discrete conservative mechanical system with infinite degrees of freedom. This implies that the free surface and velocity potential can be expressed aswhere the vector function denotes the Stokes–Zhukovsky potential, which was defined in [25], and are the generalized coordinates, the set is a Fourier basis on the unperturbed free surface, and the set should be complete in the unperturbed liquid domain. The variational procedure gives the generalized form of the infinitedimensional modal system [18], which is the complete analog of the original freeboundary problem given by equations (1)–(6) as follows:where , , , , and are the integrals over the timedependent domain and only depend on and . Their explicit formulations are given by Zhou et al. [18]. For the forced sloshing of liquid along the longitudinal axis (), equation (10) can be rewritten as
The mode shapes of the linear sloshing solutions are . According to the Moiseev asymptotic relations [25], we havewhere are the mode shapes of liquid sloshing and can be obtained by the semianalytical approach, which was developed by Wang et al. [9]. Wang et al. [9] divided the liquid domain within a baffled container into several subdomains to ensure that the liquid velocity potential for each subdomain is of class C^{1} and could be calculated with continuity boundary conditions, as shown in Figure 2. The analytical solution for the velocity potential in each subdomain can be obtained using separation of variables. The eigenfrequency equation is obtained by expanding the free surface and artificial interface conditions into the Fourier series in the direction of the liquid height and the Bessel series in the radial direction. Obviously, the partition of the liquid domain is different from that applied by Gavrilyuk et al. [32], who determined the natural frequencies and modes using the Galerkin method, which is based on Green’s function. Wang et al. [9] compared the natural frequencies obtained from these two different methods and found that the difference between the results obtained by two methods is less than 0.2%. Figure 3 shows that the natural modes calculated by Wang et al. [9] are in good agreement with those calculated by Gavrilyuk et al. [32].
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In the following study, the perturbation parameter is introduced, which denotes the ratio of the excitation amplitude and the inner radius of the container. The nonlinear modal system assumes that the forcing amplitude is sufficiently small. Miles [35, 36] demonstrated that generalized coordinates for and are dominating and are . The generalized coordinates for , , and are and the other higher modes are or . In addition, Miles [35, 36] showed that the other higher modes contribute <1% to the response. Lukovsky [31] derived the fivedimensional nonlinear modal system. The results for the surface wave height and hydrodynamic forces are in good agreement with the experimental results. The higherorder terms than will be neglected in the perturbation procedure, which means the perturbation parameter should be very small, i.e., the nonlinear analysis is considered. Substituting equation (8) in equations (9) and (11) and keeping terms up to , one has the following asymptotic modal system:where the explicit expressions of (i = 1, …, 9) have been given by [18] and σ_{mn} (m = 0, 1, 2; n = 1) denotes the natural sloshing frequencies.
3. Resultant Hydrodynamic Force and Moment
The liquid velocity potential in each subdomain has the continuous boundary conditions of class . The liquid domain with a single baffle can be divided into 4 subdomains () with 3 artificial interfaces () as shown in Figure 4, namely,
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Obviously, and are timeindependent; and are timedependent (it should be noted that and have the free surfaces and , respectively.). As shown in Figure 5, the unperturbed (hydrostatic) liquid domain can also be divided into 4 subdomains () with three artificial interfaces (), namely,where and have the unperturbed free surfaces and , respectively. The resultant hydrodynamic force of the liquid pressure acting on the container can be obtained by the formula [31]:where is the liquid mass and is the position vector with respect to the mass center of the liquid in the relative cylindrical coordinate system . The container is subjected to the lateral ground movement, and the resultant hydrodynamic force can be rewritten as
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The hydrodynamic moment about the origin due to the liquid pressure acting on the container can be obtained as follows:
In the cylindrical coordinate system , the vectors in equations (21) and (22) can be represented aswhere , , , and are the projections of the vectors on the unit vectors , , , respectively. According to equations (21) and (22), we have
The mass center of the liquid is the unique point in the liquid domain with the property that the summation of the weighted position vectors relative to this point is zero. Hence, the position vector of the mass center satisfies
From equation (24), one obtainswhere denotes the volume of the liquid.
If the surface wave height is small compared to the inner radius of the container, the integrals over can be expanded into the Taylor series about the surface wave height at the unperturbed (hydrostatic) free surface (). The Taylor expansions for () can be written as
Substituting equations (8) and (12) into (28)–(30) gives
Taking the second derivative of , we can obtain
Substituting equations (31)–(36) into (24) and (25) gives the resultant hydrodynamic force and moment, respectively. The container is subjected to the lateral ground movement along the longitudinal axis (), namely, . Owing to the symmetry of the system, the resultant hydrodynamic force along axis () and the resultant hydrodynamic moment about axis () are more significant than the other components. In the present analysis, we concentrate on the analysis of and .
4. Comparison Study
For checking the validity of the present method, the forced nonlinear sloshing of liquid in a rigid circular cylindrical container with a rigid annular baffle is studied, respectively, by using the Gavrilyuk’s modal system [33], the smoothed particle hydrodynamics (SPH), and the present method. The radius of the container is fixed at r_{2} = 1 m. The position of the unperturbed free surface is taken as z_{2} = 1 m. The container is subjected to the harmonic excitation, i.e., , where the amplitude of the container movement is X_{0} = 0.02 m. The liquid started movement from rest. According to equation (8), the initial condition for the modal system is taken as
In the first case, the baffle is positioned at z_{1} = 0.75 m and the inner radius of the baffle is taken as r_{1} = 0.4 m. Three different excitation frequencies are considered: ω = 4 rad/s, 5 rad/s, and 6 rad/s. The surface wave heights on the wall at θ = 0 obtained by the present modal system are compared with those obtained by Gavrilyuk’s modal system, as shown in Figures 6–8. It is seen from Figures 6–8 that the present results are in good agreement with those from the Gavrilyuk’s modal system.
In the second case, the baffle is positioned at z_{1} = 0.5 m and the inner radius of the baffle is r_{1} = 0.5 m. The excitation frequency is ω = 6.28 rad/s. The SPH analysis is carried out by using the commercial software ABAQUS. The surface wave profiles across the container (at θ = 0) at four different times (t = 0.5 s, 1.5 s, 2.5 s, and 3.5 s) are compared with those obtained by the SPH method, as shown in Figure 9. It is seen that good agreement is achieved between the SPH solution and the present solution.
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To verify the correctness of the present multimodal system under seismic excitation, the present results of free surface wave height are compared with the experiment results reported by Hosseinzadeh et al. [37]. A scaled steel container with 0.002 m wall thickness was used in the experiment. The diameter of the container was 1.2 m. The liquid filling height was 0.6 m. The baffle was positioned at z_{1} = 0.5 m, and the inner radius of the baffle was 0.55 m. The Tabas earthquake record (Iran, 1978), scaled to PGA = 0.4 g, was considered as the base excitation. In Figure 10, the free surface wave heights at the wall are compared with the experiment ones. As seen from Figure 8, the present results are in good agreement with the experiment ones. The typical nonlinear phenomenon can be observed from the results: the peak value of the free liquid surface is always larger than the trough value.
5. Parametric Study
The effects of baffle parameters (position and inner radius) and excitation parameters (amplitude and frequency) on nonlinear response of liquid in a container are discussed in detail. The container is subjected to a lateral harmonic excitation in the form of sinusoidal wave having the amplitude X_{0} and the frequency ω. The radius of the container is fixed at r_{2} = 1 m, and the liquid height is taken as z_{2} = 1 m.
5.1. Effect of Position of the Baffle
The inner radius of the baffle is fixed at r_{1} = 0.5 m. The amplitude of the container movement is X_{0} = 0.03 m, and the exciting frequency is ω = 6 rad/s. Under the given lateral excitation, the maximum amplitudes of nonlinear responses are presented in Table 1. f_{max} denotes the maximum amplitude of the surface wave height. F_{max} and M_{max} denote the maximum amplitudes of the resultant hydrodynamic force and moment, respectively. It is shown that f_{max} decreases as the baffle is positioned towards the free surface. In contrast, F_{max} increases with the increase of baffle position z_{1}. As the baffle moves up from the bottom, M_{max} decreases to the lowest point at z_{1} = 0.6 m. However, when the baffle further moves up to the free surface from z_{1} = 0.6 m, M_{max} tends to increase. For different baffle positions z_{1} = 0.1 m, 0.3 m, 0.5 m, and 0.7 m, the wallpressure distribution (at θ = 0) resulting in the maximum resultant hydrodynamic force and moment is shown in Figure 11. It is observed from Figure 11 that the wall pressure above the baffle decreases with the increase of z_{1}. The change of the wall pressure below the baffle versus the position of the baffle is relatively small. F_{u} and M_{u} denote the resultant hydrodynamic force and the resultant moment of the liquid above the baffle. F_{d} and M_{d} denote the resultant hydrodynamic force and the resultant moment of the liquid below the baffle. The variations of F_{u}, F_{d}, M_{u}, and M_{d} resulting in the maximum resultant hydrodynamic force and moment with respect to the position of the baffle are given in Table 2. It is seen from Table 2 that F_{u} and M_{u} decrease with the increase of z_{1} while F_{d} and M_{d} increase with the increase of z_{1}. It is seen from Table 2 that the increase rate of F_{d} exceeds the decrease rate of F_{u}. This results in the increase of the maximum resultant hydrodynamic force (F_{max} = F_{u} + F_{d}) with the increase of z_{1}. The increase rate of M_{d} is slower than the decrease rate of M_{u} when z_{1} < 0.6 m. However, the increase rate of M_{d} exceeds the decrease rate of M_{u} when z_{1} > 0.6 m. This results in the nonmonotonic variation of the maximum resultant moment (M_{max} = M_{u} + M_{d}).

 
ΔF_{u}, ΔF_{d}, ΔM_{u}, and ΔM_{d} denote the increments of the resultant force and moment from liquid below/above the baffle for the increment of the inner radius of the baffle Δz_{1} = 0.1 m. 
To investigate the effect of the baffle’s position on nonlinearity, the present nonlinear response is compared with the linear response [10, 11] within the first 30 seconds. Four different positions of the baffle are, respectively, considered: z_{1} = 0.1 m, 0.3 m, 0.5 m, and 0.7 m. The time histories of the surface wave height f_{wall} on the wall at θ = 0, the resultant hydrodynamic force F_{1} in the θ = 0 direction, and the resultant hydrodynamic moment M_{O2} about the θ = π/2 axis are depicted in Figures 12–14, respectively. It is seen from Figures 12–14 that the nonlinearity increases with the time and the nonlinearity decreases as the baffle moves close to the free surface. According to equations (1)–(6), the nonlinearity only appears on the free surface. Therefore, the nonlinearity decreases as the maximum surface wave height decreases.
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5.2. Effect of Inner Radius of the Baffle
The baffle is fixed at z_{1} = 0.5 m. The amplitude of the container movement is X_{0} = 0.03 m, and the exciting frequency is ω = 6 rad/s. The maximum amplitudes of the nonlinear response versus the inner radius of the baffle are given in Table 3. It is seen from Table 3 that f_{max} and M_{max} increase with the increase of the inner radius of the baffle. However, F_{max} decreases with the increase of baffle inner radius r_{1}. For r_{1} = 0.2 m, 0.4 m, 0.6 m, and 0.8 m, the wallpressure distribution (at θ = 0) resulting in the maximum resultant hydrodynamic force and moment is shown in Figure 15. It is observed that the wall pressure above the baffle increases with the increase of r_{1} and the wall pressure below the baffle decreases with the increase of r_{1}. The variations of F_{u}, F_{d}, M_{u}, and M_{d} with respect to the inner radius of the baffle are given in Table 4. It is seen that F_{u} and M_{d} increase with the increase of r_{1} while F_{d} decreases with the increase of r_{1}. M_{u} increases with the increase of r_{1} when r_{1} < 0.6 m. However, M_{u} decreases with the increase of r_{1} when r_{1} > 0.6 m. The variation rates of F_{d} and M_{d} are significantly faster than those of F_{u} and M_{u}, respectively.

 
ΔF_{u}, ΔF_{d}, ΔM_{u}, and ΔM_{d} denote the increments of the resultant force and moment from liquid below/above the baffle for the increment of the inner radius of the baffle Δr_{1} = 0.1 m. 
To study the effect of the baffle’s inner radius on the nonlinearity, the nonlinear response is compared with the linear response within the first 30 seconds. Four different inner radii of the baffle are considered: r_{1} = 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively. The time histories of the surface wave height f_{wall} on the wall at θ = 0, the resultant hydrodynamic force F_{1} in the θ = 0 direction, and the resultant hydrodynamic moment M_{O2} about the θ = π/2 axis are depicted in Figures 16–18. It is observed that the nonlinearity increases with the increase of the inner radius r_{1}.
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5.3. Effect of Excitation Amplitude
The inner radius of the baffle is fixed at r_{1} = 0.8 m, and the baffle is positioned at z_{1} = 0.7 m. Four different excitation amplitudes are considered: X_{0} = 0.01 m, 0.02 m, 0.03 m, and 0.04 m, respectively. The circular frequency is taken as ω = 6 rad/s. Under the lateral excitation, the maximum amplitudes of liquid response are given in Table 5. It is seen that the maximum amplitudes of liquid response increase with the increase of the excitation amplitude. To investigate the effect of the excitation amplitude on the nonlinearity of response, the present results are compared with the linear ones within the first 30 seconds. The time histories of f_{wall}, F_{1}, and M_{O2} are plotted in Figures 19–21, respectively. It is seen that the nonlinearity increases with the increase of the excitation amplitude.

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5.4. Effect of Excitation Frequency
The inner radius of the baffle is fixed at r_{1} = 0.7 m, and the baffle is positioned at z_{1} = 0.7 m. The fundamental frequency of the liquid is 3.77 rad/s. The container is undergoing the lateral harmonic excitation, and the amplitude is fixed at X_{0} = 0.02 m. Using the method developed by Faltinsen and Timokha [25], the steadystate analysis in the frequency domain for the equations (13)–(17) is conducted. Considering the harmonic excitation, the generalized coordinates for the two dominant modes can be assumed as follows:
Substituting equations (38) and (39) into (13)–(17), we havewhere satisfieswhere , , , and satisfies
The initial displacements and velocities for the free surface are assumed to be zero (i.e., the liquid is in a static state at the initial time). According to Faltinsen and Timokha [25], one can conclude the existence of the only type of motion: planar waves (i.e., ). For this case, the generalized coordinate A satisfieswhere satisfies
The response curves in the frequency domain, which are illustrated in Figure 22, can be obtained from equation (43). As shown in Figure 22, there are two branches in the plane, which have softspring behavior. According to the stability conditions for the planar waves given by Faltinsen and Timokha [25], the unstable planer waves are associated with points on the branches that are located between the two auxiliary curves () and ().
6. Conclusions
Based on the multimodal method carried out in a recent paper [18], the nonlinear response of the liquid partially filled in a rigid cylindrical container with a rigid annular baffle and subjected to lateral excitation is investigated. According to Raynovskyy and Timokha [34], the significant damping can be caused by the vortex shedding at the sharp edges of the baffles. In the subsequent study, we will quantitatively study this damping based on experimental and numerical methods and then introduce it into the multimodal system. In the present study, the surface wave height, resultant hydrodynamic force, and moment with respect to the position and inner radius of the baffle and the parameters (amplitude and frequency) of the excitation are discussed in detail. The following observations are highlighted:(1)The resultant hydrodynamic force increases as the baffle moves towards the free surface and increases with the decrease of inner radius of the baffle(2)The surface wave height decreases as the baffle is positioned towards the free surface and decreases with the decrease of inner radius of the baffle(3)The nonlinearity increases with the increase of the exciting amplitude and the accumulation of the sloshing time
Data Availability
The data used to support the findings of the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The financial supports from the National Natural Science Foundation of China (grant nos. 11702117 and 11172123), the Research Grant Council of the Hong Kong Special Administration Region, China, for the research project HKU715110E, the Jiangsu Natural Science Fund Project (BK20160482), and the Natural Science Fund (16KJB130001) for colleges and universities in Jiangsu Province are greatly appreciated.
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Copyright © 2019 Jiadong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.