Effects of excitation angle and coupled heave surge sway motion on fluid sloshing in a three-dimensional tank

Transcription

1 J Mar Sci Technol (2011) 16:22 50 DOI /s ORIGINAL ARTICLE Effects of excitation angle and coupled heave surge sway motion on fluid sloshing in a three-dimensional tank Bang-Fuh Chen Chih-Hua Wu Received: 5 January 2010 / Accepted: 7 November 2010 / Published online: 14 December 2010 Ó JASNAOE 2010 Abstract Sloshing waves in moving tanks is an important engineering problem, and most studies of this phenomenon have focused on tanks that are excited by forcing motion in a limited number of directions and with fixed excitation frequencies throughout the forcing. In practice, the excitation comprises multiple degree of freedom motion that potentially couples surge, sway, heave, pitch, roll, and yaw motions. In the present study, a time-independent finite difference method is used to simulate fluid sloshing in three-dimensional tanks filled to an arbitrary depth for various excitation frequencies and multiple degree of freedom motion. The numerical scheme developed here was verified by rigorous benchmark tests. The coupled motions of surge and sway are simulated for various excitation angles, frequencies and water depths. Five kinds of sloshing waves found under coupled surge sway motions: diagonal, single-directional, square-like, swirling, and irregular waves. The effect of excitation angle on the frequency responses of different sloshing waves is analyzed and discussed in the present study. Further, the components of horizontal force of various sloshing waves are also presented. The coupled effect of surge, sway and heave motions is also discussed, and the results show that unstable sloshing occurs when the excitation frequency of the heave motion is twice the fundamental natural frequency. Moreover, the effects of heave motion on the different types of sloshing waves are explored. It is found that heave motion causes all of the sloshing waves to change type. B.-F. Chen (&) C.-H. Wu Department of Marine Environment and Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan chenbf@mail.nsysu.edu.tw Keywords 3D tank 1 Introduction Finite difference method Sloshing fluids The phenomenon of liquid sloshing inside a partially filled tank is of great interest, and draws a great deal of attention from many researchers in the field of fluid dynamics. The physical behavior of liquids when sloshing and the factors affecting the sloshing behavior of liquids are of significant importance when designing practical and safe tanks. Under external excitations of large amplitude or when they are near the natural frequency of sloshing, the liquid inside a tank is prone to violent oscillations and can cause high impact pressure on the tank. The hydrodynamic forces exerted on the tank walls due to the sloshing of the liquid inside can damage the containers and affect the stability of moving vehicles. Abramson [1] provides a comprehensive review and discussion of early analytic and experimental studies of liquid sloshing, with applications to the aerospace industry. Neglecting fluid viscosity, a potential flow formulation of the problem was employed by Nakayama and Washizu [2], Ockendon et al. [3] and Faltinsen [4], amongst others. Besides the potential flow approaches, many numerical studies of the problem using primitive variables have been performed, particularly when the fully nonlinear effects of the waves on the free surface are included. Papers that describe the modeling of two-dimensional sloshing include those of Chen and Chiang [5], Aliabadi et al. [6], Frandsen [7], and Chen and Nokes [8]. Detailed surveys have been made by Ibrahim et al. [9], who provide general insights into sloshing problems, and by Cariou and Casella [10], who review commercial codes. The advantages and

2 J Mar Sci Technol (2011) 16: disadvantages of various computational fluid dynamics (CFD) methods are compared in the ISSC report [11] and by Ibrahim et al. [9]. Frandsen [7] developed a fully nonlinear finite difference model based on the inviscid flow equations. He described the sloshing motion in a 2-D wave tank based on potential flow theory according to a modified r-transformation that stretches the grid from the bottom to the surface. The advantage of the r-transformation is that it can avoid remeshing due to the moving free surface, and the mapping avoids the need to calculate the free surface velocity components explicitly. Moreover, free surface smoothing by means of a spatial filter is also not required. Chen and Nokes [8] studied the transient sloshing phenomenon, streamline patterns and excitation frequency effects on hydrodynamic force coefficients in detail. Wu [12] used perturbation theory to derive equations relating to the influence of the secondorder resonance of sloshing waves. This second-order resonance can occur if the sum frequency or the difference frequency of any two excitation components is equal to one of the natural frequencies. Second-order resonance can also occur when the sum of or the difference between any one of its excitation frequencies and any one of the natural frequencies is equal to another natural frequency. For example, Dx = (x e ± x i ) = x j (i, j = 1, 2, 3,, i = j), where x e is the excitation frequency of the tank, and x i and x j indicate the ith or jth natural modes, respectively. Analyses of 3D tanks are relatively rare in the literature. For 3D tank sloshing, Feng [13] used a three-dimensional marker and cell method to study fluid sloshing in a rectangular tank. This method takes large amounts of computer memory and CPU time, and the results reported indicate the presence of numerical instabilities. Wu et al. [14] used an inviscid finite element method to analyze fully nonlinear waves in a three-dimensional tank. Akyildiz and Ünal [15, 16] studied the pressure distribution in a 3D tank under the influence of baffles and for various filling levels by experiment and numerical simulation. The baffles significantly reduce the fluid motion and consequently the pressure response. Liu and Lin [17] developed a numerical model (NEWTANK) to study three-dimensional nonlinear liquid sloshing with a broken free surface. The VOF method was used and large-eddy simulation (LES) was adopted to model the effect of turbulence. The aforementioned studies focused on a tank excited in a limited range of excitation directions and used for specific applications [a liquefied natural gas (LNG) tank or liquid petroleum gas (LPG) carriers, a tuned liquid damper (TLD), an oil tank, chemical tankers], and with a fixed excitation frequency applied throughout. In reality, for a liquid tank sitting on the ground when an earthquake occurs, or for a tank floating on the sea, the excitation direction can include multiple degrees of freedom, including surge, sway, heave, pitch, roll, and yaw (see Fig. 1). This paper is an extension of the numerical investigation of Wu and Chen [25], where the liquid sloshing in a square tank was analyzed by a time-independent finite difference method under coupled surge sway motion with various excitation frequencies. In the present study, we consider the excitation of a threedimensional tank with different dimensionless excitation amplitudes and water depths; with three degrees of freedom for the surge sway and heave excitation directions; and with excitation frequencies near and far from the natural frequency. The main focus of this paper is the response of fluid in a 3-D tank undergoing horizontal and vertical combinations of motion with varying excitation directions (excitation angle: h), and the sloshing-induced forces are also considered. Section 2 introduces the equations of motion, which are written in a moving frame of reference attached to the accelerating tank. The proposed finite-difference method is introduced in Sect. 3, where the full iterative procedure is described. Section 4 presents the detailed results and provides a comprehensive discussion of all phenomena found in this study. Section 5 summarizes the key conclusions. 2 Mathematical formulation The coordinate system is chosen such that it moves with the tank (including surge, sway and heave motions), and it is illustrated in Fig. 1a. The inviscid momentum equations can be written as ou ot þ u ou ox þ v ou oy þ wou oz ¼ 1 op q ox x C ð1þ ov ot þ uov ox þ vov oy þ wov oz ¼ g 1 op q oy y C ð2þ ow ot þ uow ox þ vow oy þ wow oz ¼ 1 op q oz z C; ð3þ where u, v and w are the velocity components in the x, y and z directions, x C ; y C and z C are the acceleration components of the tank in the x, y and z directions, p is the pressure, q is the fluid density, t is the dimensional time, and g is the gravitational acceleration. The continuity equation for incompressible flow is ou ox þ ov oy þ ow oz ¼ 0; ð4þ the kinematic boundary condition on the free surface is og ot þ uog ox þ wog oz ¼ v; ð5þ where g = h(x, z, t) - d 0, d 0 is the depth of the water, and the dynamic free surface condition is

3 24 J Mar Sci Technol (2011) 16:22 50 Fig. 1 a Definition of tank motions, b the grid employed in the (x*, y*, z*) coordinate system, and c the staggered grid system p ¼ 0: ð6þ Taking partial derivatives of Eqs. 1, 2 and 3 with respect to x, y and z, respectively, and then summing the results, the following Poisson equation is obtained, which is used to solve for the pressure: o 2 p ox 2 þ o2 p oy 2 þ o2 p oz 2 ¼ q o ox uou ox þ vou oy þ wou oz q o oy uov ox þ vov oy þ wov oz q o oz uow ox þ vow oy þ wow : ð7þ oz The components of horizontal force produced by various sloshing waves are also studied, and the dimensionless force components in the x- and z-directions are evaluated as: 2 3 Z B=2 Z h Z B=2 Z h, 6 7 F x ¼ 4 p E dydz p W dydz 5 qgd0 3 F z ¼ B=2 0 Z L=2 Z h L=2 0 p N dydx B=2 0 Z L=2 Z h L=2 0 3 ; ð8þ, 7 p S dydx 5 qgd0 3 where F x and F z are the horizontal force components in x- and z-directions, respectively, p is the pressure acting on the tank walls, L is the tank length, B is the tank breadth, and the subscripts, E, W, S, and N represent integrations on the east, west, south and north walls of the tank, as shown in Fig Coordinate transformation The time-dependent free-surface boundary is transformed to a time-independent free surface in the x* y* z* domain, and no boundary tracing is needed during the calculation. A single value height function is assumed and this is evaluated by solving the kinematic free-surface condition. The irregular boundary, including the time-dependent fluid surface, the nonvertical walls and the nonhorizontal bottom, can be mapped onto a cube by the proper coordinate transformations [18, 19, 25]. The applicability of the transformations to an arbitrary sloshing tank shape might be useful when considering large-scale containers, such as harbors and dams. The shapes of these types of reservoir are arbitrary, with differently inclined walls and irregular bottoms. In this work, a square tank is used and thus the

4 J Mar Sci Technol (2011) 16: coordinate transformation equations are simplified as follows: x ¼ x b 1 b 2 b 1 y y þ dðx; zþ ¼ 1 hðx; z; tþ ð9þ ð10þ z ¼ z b 3 ; ð11þ b 4 b 3 where the instantaneous water surface, h(x, z, t), is a single-valued function measured from the tank bottom, d(x, z) represents the vertical distance between the still water surface and the tank bottom, b 1 and b 2 are horizontal distances from the x-axis to the west and east walls respectively, and b 3 and b 4 are horizontal distances from the z-axis to the north and south walls, respectively (see Fig. 1). Through Eqs. 9 11, one can map the west wall to x * = 0 and the east wall to x * = 1, the north wall to z * = 0 and the south wall to z * = 1, the free surface to y * = 0 and the tank bottom to y * = 1. In this way, the computational domain is transformed to a fixed-unit cubic domain. The main advantage of these transformations is to map a wavy and time-dependent fluid domain onto a timeindependent unit cubic domain. In this way, re-meshing due to the wavy free surface is unnecessary. In addition, the mapping implicitly deals with the free surface motion, and avoids the need to calculate the free-surface velocity components explicitly. Extrapolations are unnecessary, and free-surface smoothing by means of a spatial filter is not required. The coordinates (x*, y*, z*) can be further transformed such that the layer near the boundary is stretched to capture sharp local velocity gradients. The following exponential functions provide these stretching transformations: X ¼ k 1 þðx k 1 Þe k 1x ðx 1Þ ð12þ Y ¼ k 2 þðy k 2 Þe k 2y ðy 1Þ Z ¼ k 3 þðz k 3 Þe k 3z ðz 1Þ : ð13þ ð14þ The constants k 2 and k 2 control the mesh size near the free surface and tank bottom. The constants k 1, k 1, k 3 and k 3 similarly control the mesh in the horizontal directions. Thus, the geometry of the flow field and the meshes in the computational domain (X Y Z system) become time independent throughout the computational analysis. Note that the fluid domain in the X Y Z system remains a unit cube. In order to increase the accuracy of the computation, especially near the boundary, the stretch factors, k 1 = k 2 = k 3 = 0.5 and k 1 = k 2 = k 3 = 2, are used in the present study, and the numerical grid employed is depicted in Fig. 1b. 2.2 Dimensionless equations The dimensional parameters are normalized or nondimensionalized using the following equations: pffiffiffiffiffiffiffiffiffi U ¼ p u ffiffiffiffiffi V ¼ p v ffiffiffiffiffi W ¼ p w ffiffiffiffiffi P ¼ p gd 0 gd 0 gd 0 qgd 0 T ¼ t g=d 0 H ¼ g d 0 X c ¼ x c d 0 Y c ¼ y c d 0 Z c ¼ z c d 0 X x ¼ x d 0 Y y ¼ y d 0 Z z ¼ z d 0 ; ð15þ where x c, y c and z c are the tank displacements in the x-, y- and z-directions, respectively. With the aforementioned transformations and dimensionless variables, Eqs. 1 7 can be written in dimensionless form. For example, the momentum equation in the x-direction is U T þ C 11 C 14 U Y þ C 1 C 13 UU X þ C 2 C 14 UU Y þ C 5 C 14 VU Y þ C 8 C 14 WU Y ¼ ðc 1 C 13 P X þ C 2 C 14 P Y Þ X CTT : ð16þ The normalized forms of the other equations are listed in Appendix 1. In Eq. 16, C 1 C 15 are coefficients that arise from the coordinate transformations and can be found in Appendix 2. P X denotes a partial derivative of P with respect to X, U T is the partial derivative of U with respect to dimensionless time T, and so X CTT, Y CTT and Z CTT are dimensionless tank accelerations in the X, Y and Z directions. All other terms have similar meanings. 3 Computational algorithm In this three-dimensional analysis, a staggered grid system is used to discretize the governing equations in the computational analysis. That is, the pressure P is defined at the center of a finite difference grid cell [of dimensions (DX, DY, DZ)], whereas the velocity components U, V and W are calculated (0.5)DX, (0.5)DY and (0.5)DZ behind, above or backward of the cell center. This convention is illustrated in Fig. 1c. The Crank Nicholson second-order finite difference scheme and the Gauss Seidel point-successive over-relaxation iterative procedure are used to calculate the velocity and pressure, respectively. The numerical scheme is described below. When the dimensionless momentum equation (Eq. 16) is considered to be balanced at time T = (n? 1/2)DT, where n is the number of time steps and DT = T n?1 - T n, they can be expressed in the following finite-difference form: U nþ1 i;j;k V nþ1 i;j;k ¼ Un i;j;k DT } i;j;k þ P X i;j;k ¼ Vn i;j;k DT < i;j;k þ P Y i;j;k ð17þ ð18þ

5 26 J Mar Sci Technol (2011) 16:22 50 W nþ1 i;j;k H nþ1 i;j;k ¼ Wn i;j;k i;j;k þ P Z i;j;k ð19þ ¼ Hn i;j;k DT = i;j;k þ V i;j;k : ð20þ In these equations, the superscript n represents the time index (i.e., T = ndt). The terms without a superscript are at T = (n? 1/2)DT. The velocity components at T = (n? 1/2)DT can be approximated as the averages of the values at ndt and (n? 1)DT. All of the terms on the right-hand sides of Eqs. 17, 18 and 19 are applied at the same nodes as U i,j,k, V i,j,k and W i,j,k. The terms P X i,j,k, Y Z P i,j,k and P i,j,k are the corresponding pressure gradients in the X-, Y- and Z-directions, respectively. The term } i;j;k contain all of the remaining terms in Eq. 16, including the finite-difference expressions for the convective acceleration, and the terms related to surge, sway, and heave motions. In Eq. 20, = i;j;k is the nonlinear term. The pressure is evaluated by solving the Poisson equation, Eq. 16. For T = (n? 1/2)DT, the finite-difference equation can be expressed in the following form: P nþ1=2 i;j;k ¼ W h i P nþ1=2 i;j;k þ X nþ1=2 i;j;k þð1 WÞP i;j;k a ; ð21þ i;j;k in which a i,j,k is the sum of the coefficients of pressure P i,j,k, * W is the relaxation parameter, and P i,j,k is the previously iterated pressure. The relaxation parameter W is chosen to be 0.5 in the present study, and the iteration time is greatly reduced. The terms P i,j,k represent the finite-difference expressions of the pressure gradient and X i,j,k signify the finite-difference expressions of the nonlinear convective accelerations and the term related to tank motion (surge, sway, and heave). The detailed finite-difference expressions for P n?1/2 i,j,k, P n?1/2 i,j,k and X n?1/2 i,j,k are tedious and are therefore omitted from the text. Once the pressure field has been solved by iteration, the velocity components U n?1 i,j,k, V n?1 i,j,k and W n?1 i,j,k can be calculated from Eqs. 17, 18 and 19. The instantaneous water surface profile H n?1 i,j,k can be calculated from Eq. 20. In the present study, only the single-valued profile is assumed on the free surface. This assumption is not valid when splash or wave breaking occurs. However, it is sensitive to numerical solution for the convective term. As a result, the second-order central difference upwind scheme is used in the present numerical scheme. Using a weight parameter n, the second-order central difference and first-order upwind scheme can be written as of os ¼ 1 þ nsgnðf iþ 2 f i f i 1 Ds þ 1 nsgnðf iþ 2 f iþ1 f i ; Ds ð22þ where f can be any function of parameters such as the velocities u, v, and w, Ds stands for the spatial distance and sgn(f i ) is the positive or negative value of function f i.we also apply this scheme to solve the wave s elevation from the kinematic free-surface boundary condition. The weight parameter n is set to be 0.1 in the present numerical model as numerical damping is obviously induced by larger n. Moreover, when n is equal to 1 or 0, Eq. 22 becomes the first-order upwind or central finite difference scheme, respectively. The accuracy of the numerical results significantly depends on the spatial grid resolution and the selected time step. The numerical errors can be reduced if the time step is restricted by the condition given in Eq. 22 Dt\ min Dx min ju i;j;k j ; Dy min jv i;j;k j ; Dz min jw i;j;k j : ð23þ Equation 22 implies that a fluid particle cannot move more than one cell in a single time step. The finitedifference equations mentioned above can be used to solve for the wave field and the internal flow field, as the tank is subject to external forcing. The most difficult part of the present study is the calculation of the coefficients of pressure, a i,j,k. The detailed implicit iterative solution procedure employed here is given below. The convergence criterion is used to ensure a stable solution. In order to minimize the numerical error, the convergence criterion for the iteration of U, V, W and P is 10-5, while for H it is 10-7 in order to obtain the accurate free surface elevation. The numerical solution is obtained by implementing the following implicit iterative processes: 1. Specify the initial condition. 2. Update forcing condition (tank motion). 3. Calculate coefficients C 1 C 15 and calculate the coefficient of pressure, a i,j,k. 4. Calculate } i;j;k ; < i;j;k i;j;k : 5. Substitute the results of step 4 into Eq. 30 in order to calculate X i,j,k. 6. Using the boundary conditions on pressure, calculate the terms P Y i,j,k, P i,j,k Z and P i,j,k in order to calculate P i,j,k. 7. Calculate U i,j,k, V i,j,k and W i,j,k from Eqs. 17, 18 and 19, respectively. 8. Calculate P i,j,k from Eq. 21 and then recalculate the terms P Y i,j,k, P i,j,k and P i,j,k. 9. Recalculate new U i,j,k, V i,j,k and W i,j,k from Eqs. 17, 18 and 19, respectively. 10. Repeat steps 6 9 at least 2 times, then check for convergence; the convergence criteria are: P k?1 - P k \ 10-5, U k?1 - U k \ 10-5, V k?1 - V k \ 10-5 and W k?1 - W k \ 10-5, in which k represents the iteration number. If convergence is not reached, repeat steps 4 10.

6 J Mar Sci Technol (2011) 16: Calculate H i,j,k from Eq. 20 and check that H k?1 - H k \ If convergence has not been reached, go to step 3 and update the coefficients relating to H. 12. If H has converged then go to step 2 and begin the next time step. 4 Results and discussion The wave motion in a three-dimensional tank is considerably more complicated than that in a two-dimensional tank. For a three-dimensional tank traveling in a vehicle on a highway, or on a ship on the sea, the angle of the excitation motion can change randomly over time, and a small perturbation in excitation angle might significantly affect the sloshing response of the fluid in the tank. As indicated earlier, if the tank is excited by an earthquake, the duration of which may be less than 60 s, then a steady-state response is unlikely to be achieved during the time span of the excitation, so the transient motion will be the most important. Therefore, our main emphasis in this study is to focus on the transient fluid motion caused by harmonic excitation with various excitation frequencies and angles. The natural modes of a 3-D tank with a square base can be obtained by solving the linearized natural sloshing standing-wave problem. The angular frequencies k i,j of these natural modes are given in Eq. 24: p k i;j ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 þ j 2 x 2 i;j ¼ gk ; ð24þ i;j tanh k i;j d 0 where i, j are the natural mode s components along the x- and z-axes, and x 2 i,j are the natural frequencies of the 3-D tank. 4.1 Benchmark tests To validate the accuracy of our model, the simulated results are compared with those reported in the literature. Figure 2a shows the sloshing displacement in the corners A and B of a tank with a depth-to-breadth ratio d 0 /L = 0.25 when the tank is diagonally excited by a ground motion with an excitation displacement a 0 = L and an excitation frequency equal to 0.99x 1 (x 1 = x 1,0, the first natural frequency of the tank). The results reported by Kim [20] are also shown in the figure, and the agreement is satisfactory. Kim used the SURF scheme to solve for the kinematic boundary condition, and used a five-point numerical filtering formula to avoid instability due to sawtooth waves. We use simple mapping functions to remove the time dependence of the free surface. In this way, re-meshing due to the wavy free surface is unnecessary and the need to calculate the free-surface components explicitly is avoided. Besides, the present numerical scheme can achieve accurate numerical results based on relatively coarse meshes (X 9 Y 9 Z = ) compared with Kim s ( ) meshes. Both simulation time and memory are, therefore, dramatically reduced in this study. For instance, the computational time for coarse grid numbers (X 9 Y 9 Z = ) is about 3.5 times less than that for finer grid numbers ( ) when implementing the present numerical simulation. In addition, the corresponding memory of the coarse grid ( ) is reduced by a factor of 40 compared to that of finer grid numbers. Similar to parametric study [25], a grid resolution of DX = DZ = 0.05, DY = 0.1 and a time step DT = are used in all of the simulations in the present study. The other comparison is shown in Fig. 2b, which illustrates a tank under heave motion with an initial perturbation of the free surface. A tank with d 0 /L = 0.5, a 0 /L = and a vertical excitation frequency x y = 2.0x 1 was simulated, and the agreement was also good. Figure 3 compares the present numerical results with experimental measurements and analytical results reported by Faltinsen et al. [21]. Both longitudinal (Fig. 3a, b) and diagonal (Fig. 3c, d) forcing are implemented for a square tank with d 0 /L = 0.5, a 0 /L = The corresponding Fig. 2 Comparison between the present results and the reported numerical results. a Diagonal motion, the wave history at corner A (solid line) and B (dash line); ratio d 0 (water depth)/l = 0.25, ratio of excitation displacement a 0 /L = , x x = x z = 0.99x 1. b Heave motion; ratio d 0 /L = 0.5, a 0 /L = 0.026, x y = 2.0x 1, t real time, a h the initial perturbation = m (Frandsen 2004) a b h / a h the present result Frandsen (2004) t x ω 1

7 28 J Mar Sci Technol (2011) 16:22 50 Fig. 3 Comparison of the present results with the experimental and theoretical results reported by Faltinsen et al. (2005), d 0 /L = d 0 /B = 0.5. Excitation displacement a 0 /L = ; a, b surge motion, x x = 1.037x 1 ; c, d diagonal motion, x x = x z = 1.115x 1 Fig. 4 Comparison between the present numerical results and the V21 case of Bredmose et al. [26]. d 0 /L = 0.21, a 0 / L = , x x = 2.26x 1, y 0 /L = , x y = 2x x longitudinal force (F x ) is also presented. Furthermore, a comparison between the present numerical results and those of Bredmose et al. [26] is given in Fig. 4 for a tank initially excited by sway motion (t = s) and successively excited by heave motion (t = s). The comparisons mentioned above show, once again, excellent agreement among the numerical, experimental and analytical results. 4.2 Horizontal ground motion For practical engineering applications, if we focus on the horizontal excitation only, the tank would generally not be excited in either the surge (x-) or the sway (z-) direction; rather, it will be a coupled surge-and-sway motion, and the excitation direction may vary with time. In this paper, we define h as the excitation angle of the horizontal ground motion (see Fig. 1a). In the present study, longitudinal excitation and diagonal excitation with a wide range of excitation frequencies are simulated. The tank is rigid with a square base, and is filled to various fluid depths of d 0 /L = 0.15, 0.25, 0.5 and 1.0. The amplitude of the ground displacement is 0.005L. The different types of sloshing wave that arise with different excitation frequencies and excitation angles are discussed separately in the following subsections Single-directional and square-like waves For nonresonant excitation, a single-directional wave indicates that the wave sloshes only in the same direction as the tank s excitation. Figure 5 shows the sloshing displacement of the fluid in a tank excited at various excitation angles and at a constant excitation frequency x = 0.4x 1. Figure 5a, b illustrate the wave histories at points E and F. The sloshing amplitude at point E increases

8 J Mar Sci Technol (2011) 16: Fig. 5 The effect of various excitation angles for diagonal or single-directional waves. The excitation frequency x x = x z = 0.4x 1. a The wave history of point F, b the wave history of point E, and c the pattern of diagonal and singledirectional waves Fig. 6 The longitudinal (F x ; a) and transverse (F z ; b) forces of single-directional waves for various excitation angles. x x = x z = 0.4x 1. Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 with the excitation angle, while the response at point F shows the opposite trend. The reason is trivial. Point E is in the direction of sway and the displacement in the sway direction increases with the excitation angle. Under this excitation frequency, the sloshing waves remain as singledirection waves. Figure 5c depicts the relationship between the elevations at points E and F, and the patterns vary with different excitation angles. The corresponding components of horizontal force on the tank walls are illustrated in Fig. 6. The influence of excitation angle on the longitudinal force (F x ) looks unclear when h \ 30, as the component of excitation in the x-direction is a cosine function; as a result of the insignificant variation of the cosine function when 0 \ h \ 30, F x seems to be independent of the direction of movement of the tank when the angle of excitation is smaller than h = 30. On the other hand, an obvious influence of the excitation angle on the transverse force (F z ) is observed and is presented in Fig. 6b. The phenomenon of square-like waves corresponds to waves traveling primarily on two opposite sides of the tank. Figure 7 plots the results for square-like waves generated by a coupled surge-and-sway motion with an excitation frequency of 1.5x 1. Figure 7a, b illustrates the corresponding wave histories at points E and F and show a similar behavior to that in Fig. 5a, b. The relationship between sloshing displacements at points E and F are shown in Fig. 7c g for five different excitation angles, and the plots show only minor differences. The corresponding free-surface contours are shown in Fig. 8, which clearly demonstrates that distinct terrace areas appear with various excitation angles. The components of horizontal force of square-like waves under different excitation angles are

9 30 J Mar Sci Technol (2011) 16:22 50 Fig. 7 The effects of various excitation angles for square-like waves. The excitation frequency x x = x z = 1.5x 1. a The wave history of point F, b the wave history of point E, c g patterns of square-like waves. Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 presented in Fig. 9. The influence of excitation angle on square-like waves is similar to the influence of excitation angles on single-directional waves Irregular waves Waves that slosh irregularly inside the tank are termed irregular or chaotic waves. Figure 10 shows the sloshing responses of irregular waves. Figure 10a, b illustrate the wave histories of points E and F, and these histories are quite different from those of the other kinds of sloshing waves. The relationships between sloshing displacements at points E and F of the tank excited by various excitation angles are plotted in Fig. 10c h. The free-surface displacement at point E apparently exhibits a chaotic response to changing excitation angle, whereas Fig. 10a shows a response at point F that is almost independent of excitation angle, except for h = 45. More precisely, the sloshing

10 J Mar Sci Technol (2011) 16: Fig. 8 The free-surface contour profiles of square-like waves for various excitation angles. The excitation frequency of the tank is 1.5x 1. a h = 5, b h = 10, c h = 15, d h = 30, e h = 45 Fig. 9 The longitudinal (F x ; a) and transverse (F z ; b) forces of square-like waves for various excitation angles. x x = x z = 1.5x 1. Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 displacements at point F, excited by various excitation angles, are almost the same as that of the tank excited by surge motion only. The corresponding components of horizontal force are depicted in Fig. 11, and the excitation angles have a very small influence on the longitudinal force (F x ) but an obvious effect on the transverse force (F z ). This may be explained by the spectral analysis for irregular waves. As can be seen in Figs. 12 and 13, spectral analyses of the sloshing histories of the elevation at point F show almost identical amplitudes for the resonant peaks that correspond to the odd natural modes (x 1,0, x 3,0 and x 5,0 ) of the tank system. The mode shapes of the odd modes are shown in Fig. 14, and the sloshing displacement along the x-axis may be attributed to the superposition of the above three modes. Since point F is on the west wall normal to the x-direction (the surge direction), varying the

11 32 J Mar Sci Technol (2011) 16:22 50 Fig. 10 The effects of various excitation angles for irregular waves. The excitation frequency x x = x z = 2.3x 1. a The wave history of point F, b the wave history of point E, c h patterns of irregular excitation angle will change the excitation amplitude in the surge direction, but the dominant influence of the odd modes will generate the same order of sloshing displacements at point F (see Figs. 12, 13). Spectral analyses of the sloshing displacement at point E (see Figs. 12, 13) show that the resonant peaks vary with the excitation angle, and the sloshing displacement at point E is quite dependent on the excitation angle. The excitation amplitude in the z-direction (sway direction) increases with excitation angle, and point E is on the south wall normal to the z-direction waves. Stars h = 0. Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 (sway direction). Therefore, the sloshing displacement at point E increases with the excitation angle Swirling waves Swirling-like waves, which mean that the wave traves inside the tank within the limitations of a swirling route, occur as the tank is excited at a frequency below or above the excitation frequency of the swirling waves. This type of wave was not reported in the literature, because all

12 J Mar Sci Technol (2011) 16: Fig. 11 The longitudinal (F x ; a) and transverse (F z ; b) forces of irregular waves under various excitation angles. x x = x z = 2.3x 1. Stars h = 0. Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 Fig. 12 Power spectral analyses of irregular waves at point E and point F with various excitation angles, x x = x z = 1.8x 1. a h = 0, 5, 10, point E; b h = 15, 30, 45, point E; c h = 0, 5, 10, point F; d h = 15, 30, 45, point F

13 34 J Mar Sci Technol (2011) 16:22 50 Fig. 13 Power spectral analyses of irregular waves at point E and point F with various excitation angles, x x = x z = 2.3x 1. a h = 0, 5, 10, point E; b h = 15, 30, 45, point E; c h = 0, 5, 10, point F; d h = 15, 30, 45, point F previous studies have been limited to a tank undergoing longitudinal forcing (surge h = 0 ) or diagonal forcing (h = 45 ). The wave histories at points E and F for swirling waves are shown in Fig. 15a, b. Under near-resonant excitation, the occurrence of the swirling is clearly affected by the excitation angle of the horizontal ground motion. Figure 15c g presents the relationship between the sloshing displacements at points E and F for a tank excited at various excitation angles. For the case of h = 45, the sloshing displacement at the corners of the tank (points A and C) ultimately become greater than twice the water depth, d 0, and unrealistic wave heights occur after T = 110, so the simulations are stopped at T = 110. As shown in Fig. 15c g, the swirling phenomena for the cases h = 5, 10 and 15 are obvious, and both counterclockwise (solid line) and clockwise (dashed line) swirling are present. The swirling behavior is small when h is larger than 30, and it has essentially disappeared by h = 45, implying that the swirling waves are very difficult to generate when the tank is excited by near-diagonal forcing.

14 J Mar Sci Technol (2011) 16: Fig. 14 Free surface profiles of natural modes. a x i,j = x 1,0 ; b x i,j = x 3,0 ; c x i,j = x 5,0 The corresponding sloshing-induced forces are illustrated in Fig. 16, and the components of horizontal force are mainly dominated by the sloshing displacement. However, for a longer time simulation (T is up to 800), a significant influence of the excitation angle on the rotational directions of the swirling waves can be elucidated (as shown in Fig. 16c, d), which causes different beating periods for the swirling waves. Thus, various amplitudes and periods of forces are observed. The beating periods of the longitudinal force (F x ) seem to be independent of the excitation angle when h B 15. The beating periods of the transverse force (F z ) are much larger than those of the longitudinal force, and they decrease as the excitation angle increases. Although the swirling wave switches its swirling direction alternately, the change in excitation direction generates a variety of swirling waves such that they all have dissimilar sloshing displacements, horizontal forces and beating periods Classification of sloshing waves Faltinsen et al. [22] provided an extensive experimental dataset that they were able to compare with the results from their adaptive asymptotic theory. They classified the sloshing waves into four categories: planar, diagonal, chaotic, and swirling waves. Although the transient responses were measured in their experimental investigation, the steady-state responses provide the major focus of most of their published articles. The excitation frequencies used in their studies were limited to the range x 1, and near primary resonance excitation was investigated in detail. In this study, simulations have been performed under a broader range of excitation frequencies and excitation angles. We extended the range of excitation frequencies to x 1, and in addition to longitudinal forcing and diagonal forcing, four different excitation angles were analyzed. The classification of the sloshing waves in the present study, for a tank excited in horizontal motion with various excitation angles and frequencies, is illustrated in Fig. 17. In the paper of Faltinsen et al. [23], the chaotic waves predicted by their analytical model and experiments were found to occur within a small excitation frequency range close to the first resonant mode. For an excitation frequency within the same range as that used in the studies of Faltinsen et al., chaotic waves were successfully simulated in the present study, even in the diagonal excitation case. Chaotic waves were also found in other simulations where the excitation frequency is away from the first fundamental mode. The range of excitation frequencies in which the swirling waves are observed is x 1 when the excitation angle is less than 30. The range of excitation frequencies for swirling waves shrinks to a very narrow band when the tank is under diagonal forcing. This classification is in very good agreement with that of Faltinsen and coauthors. We find from Fig. 17 that one kind of sloshing wave, which we name swirling-like (SWL) waves, appears within a certain range of excitation frequencies. These swirling-like (SWL) waves occur before and after the range of excitation frequencies of swirling waves, and they swirl within a limited domain inside the tank. The historical distribution of absolute peaks of swirling-like (SWL) waves is shown in Fig. 18. A more detailed discussion of this and the effect of heave motion on SWL waves will be provided in Sect The classifications of sloshing waves for various water depths were also studied and are illustrated in Figs. 19, 20,

15 36 J Mar Sci Technol (2011) 16:22 50 Fig. 15 The effects of various excitation angles for swirling waves. The excitation frequency x x = x z = 0.97x 1. a The wave history of point F; b the wave history of point E; c g patterns of swirling waves. and 21. For a shallow water depth (d 0 /L = 0.15), as shown in Fig. 19, there is an obvious difference in the distribution of wave types compared to those of d 0 /L = The swirling waves not only appear within a wider frequency range ( x 1 ) but they coexist with the other types of sloshing waves (e.g., irregular or square-like waves). This may be related to the effect of nonlinearity of sloshing displacement, which causes more resonant modes to trigger the generation of swirling waves. In contrast, the frequency domain of swirling waves shrinks with increasing water depth (see Figs. 20, 21), and we failed to predict swirling waves at finite depth (d 0 /L = 0.5 and 1.0) when the excitation angle is close to 45. Moreover, the frequency domain of swirling-like waves increases with increasing water depth, and for the frequency domain of square-like waves, it varies with water depth and mingles with the Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 other types of sloshing waves. When the ratio d 0 /L = 0.15, square-like waves are found to mingle with singledirectional waves (x/x 1 *0.7) and swirling waves (x/x 1 *1.3). In addition, square-like waves and irregular waves coexist at finite depth when the ratio x/x 1 is close to 1.5. The frequencies of the irregular waves widen with water depth. This may be associated with the influence of higher natural modes when the external forcing frequency is close to odd natural modes at finite depth (for example, d 0 /L = 1.0, x 3,0 = 1.73x 1 ). Power spectral analyses of various sloshing waves for different water depths are discussed in detail here. For single-direction waves, as shown in Fig. 22, the dominant frequency is the external forcing frequency, and the secondary resonant mode is the first natural frequency. The influence of water depth is insignificant on single-direction

16 J Mar Sci Technol (2011) 16: Fig. 16 Longitudinal (F x ; a, c) and transverse (F z ; b, d) forces of swirling waves under various excitation angles. x x = x z = 0.97x 1. Continuous line h = 5, dashed line h = 10, dotted line h = 15, dashed dotted line h = 30, straight line with multiplication symbol h = 45 Fig. 17 The classification of sloshing waves of surge sway motion. d 0 /L = d 0 /B = 0.25, a 0 /L = D diagonal waves, S swirling waves, I irregular waves, SL square-like waves, SWL swirling-like waves, SD single-directional waves waves. When it comes to square-like waves (see Fig. 23), except for the fundamental natural frequency (x 1 ) and the excitation frequency (1.5x 1 ), the third mode (x 3,0 ) of the natural frequency appears when d 0 /L [ This may stem from the forcing frequency (1.5x 1 ), which is much closer to x 3,0 in the finite depth, and so the square-like waves coexist with the irregular waves at this forcing frequency. As shown in Fig. 24, double peaks occur in the power spectral analyses of the swirling waves around the first natural frequency. The dominant frequencies of the

17 38 J Mar Sci Technol (2011) 16:22 50 swirling waves are influenced by the effect of excitation angles, especially when d 0 /L = 1.0. This indicates that the main resonant modes of swirling waves are shifted a little to the right with increasing excitation angle (see Fig. 24). Further, the secondary resonant modes (2x 1,0,3x 1,0,2x, 3x) can be clearly seen in Fig. 24. However, the second resonant modes seem to disappear, as the ratio d 0 /L is over 0.5. In other words, the strong nonlinear effect of sloshing waves can cause the occurrence of secondary resonance at shallow water depths. Thus, the association between the nonlinearity of the sloshing waves and the secondary resonance may be the key to triggering the generation of swirling waves, particularly for shallow water depths. For irregular waves (see Fig. 25), the excitation frequency of the tank coexists with the other resonant modes in the power spectral analyses, indicating the significant influence of the odd natural modes on irregular waves in particular [25]. 4.3 Coupling surge sway and heave excitations Surge sway heave motion Fig. 18 The distribution of absolute peaks for swirling-like waves. x x = x z = 0.9x 1, h = 5 The heave motion is just a change in gravitational acceleration, and its effect is expected to be small if the free surface of the tank is initially undisturbed. However, if the free surface is initially inclined, or it is disturbed by other combinations of ground motion, the vertical excitation is likely to enlarge the free surface elevation during tank motion. The detailed relations between an initially disturbed or planar free surface and heave motion are reported in Chen and Nokes [8] for a two-dimensional tank. The linear solution for the motion of fluid in a vertically excited tank was first obtained by Benjamin and Ursell [24], and the stability of heave motion was included in their Fig. 19 The classification of sloshing waves of surge sway motion. d 0 /L = d 0 /B = 0.15, a 0 /L = D diagonal waves, S swirling waves, I irregular waves, SL square-like waves, SWL swirling-like waves, SD single-directional waves Fig. 20 The classification of sloshing waves of surge sway motion. d 0 /L = d 0 /B = 0.5, a 0 /L = D diagonal waves, S swirling waves, I irregular waves, SL square-like waves, SWL swirling-like waves, SD single-directional waves Fig. 21 The classification of sloshing waves of surge sway motion. d 0 /L = d 0 /B = 1.0, a 0 /L = D diagonal waves, S swirling waves, I irregular waves, SL square-like waves, SWL swirling-like waves, SD single-directional waves

18 J Mar Sci Technol (2011) 16: Fig. 22 Power spectral analyses of single-directional waves and diagonal waves with various excitation angles and water depths. a d 0 /L = 0.15, x x = x z = 0.7x 1 ; b d 0 /L = 0.25, x x = x z = 0.4x 1 ; c d 0 /L = 0.5, x x = x z = 0.4x 1 ; d d 0 /L = 1.0, x x = x z = 0.4x 1 study. Frandsen [7] considered the homogeneous and nonhomogeneous Mathieu equations in order to solve for the stable and unstable regions of a tank under vertical motions. For a tank under heave motion with a disturbed free surface and a vertical excitation frequency equal to or near 2x 1, the destabilizing influence of the heave motion is significant. In the present study, we investigate the effect of heave motion with an excitation frequency of x y = 2x 1 coupled with horizontal ground motions on the different sloshing waves described above. The excitation angle is 5. The components of horizontal force are also investigated. As suggested in Chen and Nokes [8], the time at which the heave motion begins may affect the sloshing displacement of the fluid in the tank. Figure 26 shows the time histories of the sloshing displacement of fluid in a tank with an initially horizontal surface the dotted line corresponds to the heave motion starting at T = 0, while the solid line corresponds to a start time of T = 63 (t = 10 s). The vertical excitation displacement y 0 of the tank is 0.01L and the excitation frequency of heave motion is 2.0x 1. As shown in the figure, the resulting sloshing displacements in the former case are significantly larger than those in the latter case. In the former case, the effect of heave motion on sloshing displacement is not apparent until T [ 30. On the other hand, for the latter case, the heave motion is included in the excitation at T = 63; before then, the free surface of the fluid in the tank has been disturbed by the surge sway motion, and the effect of heave motion on sloshing displacement is excepted to be larger than in the former case. As can be seen in Fig. 26, the effect of heave motion is obvious after T = 80, which implies that a disturbed free surface can be more easily triggered by the heave motion. Ultimately, the sloshing displacements moved beyond the limit of the present numerical model and the simulations were halted. Figure 27 shows the results for a tank excited by coupled surge sway heave motion with various vertical excitation displacements. The free surface in the tank is initially undisturbed, and the heave motion starts at T = 63 in all cases. The figure shows that the destabilizing influence of the heave motion is clearly noticeable for the case with the largest amplitude, and violent sloshing starts much earlier than it does for a tank under surge sway excitation only. The simulation results diverged at T = 130 and 300 for y 0 /L = 0.01 and y 0 /L = 0.005, respectively. For the case with the smallest vertical excitation amplitude, y 0 /L = 0.002, the influence of heave motion is not obvious. Thus, in the following sections, the vertical displacement (y 0 ) was chosen to be 0.005L, with heave motion starting at T = 63, in all of the simulated cases.

19 40 J Mar Sci Technol (2011) 16:22 50 Fig. 23 Power spectral analyses of square-like waves with various excitation angles and water depths. a d 0 / L = 0.15, x x = x z = 1.5x 1 ; b d 0 /L = 0.25, x x = x z = 1.5x 1 ; c d 0 / L = 0.5, x x = x z = 1.5x 1 ; d d 0 /L = 1.0, x x = x z = 1.5x 1 Fig. 24 Power spectral analyses of swirling waves with various excitation angles and water depths. a d 0 /L = 0.15, x x = x z = 0.99x 1 ; b d 0 / L = 0.25, x x = x z = 0.97x 1 ; c d 0 /L = 0.5, x x = x z = 1.04x 1 ; d d 0 / L = 1.0, x x = x z = 1.03x 1

20 J Mar Sci Technol (2011) 16: Fig. 25 Power spectral analyses of irregular waves with various excitation angles and water depths. a d 0 /L = 0.15, x x = x z = 2.0x 1 ; b d 0 / L = 0.25,x x = x z = 2.3x 1 ; c d 0 /L = 0.5, x x = x z = 2.0x 1 ; d d 0 / L = 1.0, x x = x z = 1.8x 1 Fig. 26 The wave elevation results for corner A under coupled surge sway heave motion. Solid line heave motion starts at T = 63, dotted line heave motion starts at T = 0. d 0 /L = 0.25, x x = x z = 0.97x 1, h = 5, y 0 = 0.01L, x y = 2.0x Effect of heave motion on single-directional or diagonal waves Figure 28 depicts the effect of heave motion on singledirectional waves. As shown in Fig. 28a, the influence of the heave motion becomes significant at T = 800. The sloshing wave pattern switches from single-directional waves to counterclockwise-swirling waves once the heave motion begins to have an effect, as can be seen in Fig. 28a, b. A spectral analysis (Fig. 28c, d) exhibits two peaks corresponding to 0.5x 1, the excitation frequency, and 1.0x 1, the first fundamental frequency early in the time Fig. 27 Wave history of corner A as a tank under coupled surge sway heave motion with various vertical excitation displacements. Heave motion starts at T = 63x x = x z = 0.97x 1, h = 5, x y = 2.0x 1 history (T \ 470). After T = 600, as the influence of the heave motion gradually becomes apparent, the spectral analysis shows rapid peak growth at the first natural frequency and the appearance of a second peak at 2.0x 1.At these later times, the magnitude of the peak corresponding to the excitation frequency is insignificant. As stated in the previous section, the resonant peak corresponding to x 2,2 = 2.05x 1 may be related to the occurrence of swirling

21 42 J Mar Sci Technol (2011) 16:22 50 Fig. 28 Effect of heave motion on single-directional waves. a Wave pattern (the figure in the top right corner shows the wave history of point A), b distribution of peaks, c spectral analysis at T = 0 470, d spectral analysis at T = x x = x z = 0.5x 1, h = 5o, x y = 2.0x 1, y 0 = 0.005L Fig. 29 Effect of heave motion on diagonal waves. a Wave pattern (the figure in the bottom left corner is the wave history of point A), b distribution of peaks, c spectral analysis at T = 0 470, d spectral analysis at T = x x = x z = 0.5x 1, h = 45, x y = 2.0x 1, y 0 = 0.005L waves [25]. Thus, the addition of heave motion to the coupled surge sway motion can generate resonant sloshing at a frequency close to x 2,2 and change the sloshing pattern from single-direction waves to swirling waves. The effect of heave motion on the diagonal waves can be seen in Fig. 29. The heave motion starts to have an obvious influence after T = 700, but it is clear from Fig. 28a and b that in this case the wave pattern does not

22 J Mar Sci Technol (2011) 16: Fig. 30 Force components of a single-directional waves (h = 5 ) and b diagonal waves (h = 45 ) under coupled surge sway heave motion. x x = x z = 0.5x 1, x y = 2.0x 1 switch to a swirling wave pattern during the duration of the simulation. As stated before, swirling waves are seldom found in a tank under diagonal forcing; however, even though the simulation time is limited, the spectral analysis of the sloshing history again indicates the presence of a peak corresponding to 2.0x 1, and it may be that the introduction of heave motion ultimately leads to the generation of swirling waves in the tank, even under diagonal forcing. The components of horizontal force for the singledirectional and diagonal waves are depicted in Fig. 30a and b, respectively. As shown in Fig. 30a, the heave motion starts to have a noticeable influence on the longitudinal force (F x ) earlier that it does on the transverse force (F z ) for single-directional waves. F x is excited by heave motion at T*800, after which it increases rapidly with time. The stimulation of F z by the heave motion is very apparent after T = 1000, and F z increases very quickly between T = 1100 and F x and F z end up with almost identical values, and the simulation is stopped due to the limits of the present numerical model. For the diagonal wave results shown in Fig. 30b, F x and F z present nearly the same amplifications under the influence of heave motion Effect of heave motion on square-like waves Figure 31 shows the results for square-like waves affected by heave motion, and again a delayed influence is apparent beginning around T = 600. At this point, the sloshing displacement increases dramatically and the original square-like waves turn into swirling waves after T = 700. Figure 31a presents the wave pattern for surge sway heave motion (the wave pattern of surge sway motion is shown in the top right-hand corner), and the square-like waves change to counterclockwise-swirling waves. Figure 31b shows the peak distribution for surge sway heave motion (the peak distribution for surge sway motion is shown in the top right-hand corner), which further confirms the occurrence of a swirling pattern when the tank is subjected to coupled surge sway heave motion. Figure 31c shows that, initially, the dominant frequencies are the primary resonance frequency and the excitation frequency 0.8x 1. However, as the effect of heave motion increases, the dominant frequency becomes the first natural frequency x 1, with a secondary peak occurring at 2x 1,as seen in Fig. 31d. The appearance of this peak again signals the generation of swirling waves. The corresponding horizontal force components are illustrated in Fig. 32, and the effect of heave motion is not very apparent before T = 700. As the effect of heave motion becomes more significant, F z increases faster than F x, and they become almost equivalent to each other in the end as the program diverges Effect of heave motion on swirling-like waves For an excitation angle h = 5, swirling-like waves occur when the excitation frequencies are within the range x = x 1. For the swirling wave pattern, the absolute peaks are scattered along all the tank walls, while the absolute peak distribution for swirling-like waves shows that the peaks are basically scattered along the tank walls AB and CD (see Fig. 18). The highest sloshing displacements clearly occur in the corner in the case of swirling waves (see Fig. 25), but the location of the highest sloshing displacements is not obvious for swirling-like waves. The effect of heave motion on swirling-like waves is illustrated in Fig. 33, and Fig. 33a provides a comparison between the free surface motion at point A when the tank is excited by surge sway motion only and when it is excited by heave surge sway motion. Beginning at around T = 420, the effect of heave motion is substantial. Figure 33b, c indicate that the heave motion eventually converts the swirling-like motion into real swirling waves. Figure 34 depicts the repeated switching of the direction of swirling a phenomenon that does not occur when the tank is excited by horizontal motion with an excitation frequency of 0.9x 1. This switch in swirling direction only occurs when the tank

23 44 J Mar Sci Technol (2011) 16:22 50 Fig. 31 Effect of heave motion on square-like waves. a Wave pattern (the figure in the top right corner is that for surge sway motion alone), b distribution of peaks (the figure in the top right corner is that for surge sway motion alone), c spectral analysis at T = 0 380, d spectral analysis at T = x x = x z = 0.8x 1, h = 5, x y = 2.0x 1 Fig. 32 Components of the forces of square-like waves (x x = x z = 0.8x 1 ) under coupled surge sway heave motion. h = 5, x y = 2.0x 1 is excited by surge sway motion with an excitation frequency that is close to the first mode. The time durations of clockwise and counterclockwise swirling are shorter when the heave excitation is included in the tank motion. A spectral analysis of the sloshing displacement of fluid in a tank excited by coupled heave surge sway motion is shown in Fig. 34b and c. Two peaks corresponding to the forcing frequency, 0.9x 1, and the natural frequency, 1.0x 1, occur when T = 0 400, and three peaks (0.9x 1, 1.0x 1, and 2.0x 1 ) appear during T = Although the dominant peak is 1.0x 1, once again, the resonance corresponding to 2.0x 1 is related to the occurrence of swirling waves. The effect of heave motion on the swirling-like waves is obvious, and the heave motion will increase the switching frequency and may lead to instability of vehicles carrying liquid tanks. Figure 35 presents the corresponding longitudinal and transverse forces, and once again the salient effect of heave motion on the horizontal force is demonstrated. Violent clockwise-swirling waves occur after T = 500, and in the meantime, the transverse force (F z ) increases rapidly with an inverse phase that is comparable to that of F x. Just like the other sloshing waves under the influence of heave motion, strong clockwiseswirling waves with large sloshing displacements and horizontal force components move beyond the limits of the numerical scheme.

24 J Mar Sci Technol (2011) 16: Fig. 33 Effect of heave motion on swirling-like waves, x x = x z = 0.9x 1, h = 5, x y = 2.0x 1. a Wave history of point A, b wave pattern (the figure in the top right corner is that for surge sway motion only), c distribution of peaks (the figure in the top right corner is that for only surge sway motion only) Fig. 34 Time regions of clockwise- and counterclockwise-swirling waves and a spectral analysis of wave elevation under coupled surge sway heave motion. x x = x z = 0.9x 1, h = 5, x y = 2.0x 1. a Time regions of clockwise- (solid double-headed arrows) and counterclockwise- (dotted double-headed arrows) swirling waves, b spectral analyses of points A and B at T = 0 400, c spectral analyses of points A and B at T = Effect of heave motion on swirling waves The sloshing motions for a tank with swirling waves excited by ground motion with and without heave motion are plotted in Fig. 36. Figure 36a demonstrates that violent sloshing becomes apparent after T = 150, when the heave motion is added to the tank excitation, and it is even more evident in the second beating period. In contrast to the effect of heave motion on swirling-like waves, the effect of heave motion on the swirling waves diminishes the phenomenon where the swirling switches direction, and the direction of the swirling wave ultimately becomes constantly counterclockwise. The reason for this can be deduced from the sloshing displacement of fluid at the four

25 46 J Mar Sci Technol (2011) 16:22 50 Fig. 35 Components of the forces of swirling-like waves (x x = x z = 0.9x 1 ) under coupled surge sway heave motion. h = 5 x y = 2.0x 1 Fig. 36 Effect of heave motion on swirling waves, x x = x z = 0.97x 1, h = 5, x y = 2.0x 1. a The wave history of point A, b wave pattern (the figure in the top right corner is that for surge sway motion only), c distribution of peaks (the figure in the top right corner is that for surge sway motion only), d spectral analysis of point A at T = 0 150, e spectral analysis of point A at T = corners of the tank, as shown in Fig. 36c, d. The wave elevation histories at points A and B are similar to those at points C and D, and this phenomenon is the same as that for a tank excited by a surge motion only. The effect of heave motion may be to cause a significantly increased sloshing displacement, and hence enlarge the contribution of the surge excitation to the system, so the swirling direction stops switching. Spectral analysis of the waves

26 J Mar Sci Technol (2011) 16: further indicates that the resonant peaks are at 1.0x 1 and 2.0x 1 during T = 0 150, and that those peaks become 0.99x 1, 1.98x 1, and 2.03x 1 for T = This minor shift in the highest resonant frequency could affect the entire sloshing phenomenon. The components of the horizontal force of the swirling waves under coupled surge sway heave motion are Fig. 37 The components of the forces of swirling waves (x x = x z = 0.97x 1 ) under coupled surge sway heave motion. h = 5 delineated in Fig. 37. F x and F z are almost in phase until T = 100, and the phase lag between F x and F z is associated with not only the occurrence but the rotational direction of the swirling waves. A near-identical phase lag between F x and F z, therefore, results in the presentation of sustained counterclockwise-swirling waves Effect of heave motion on irregular waves Figure 38 illustrates the effect of heave motion on irregular waves. It appears from Fig. 38a that the influence of heave motion is significant after T = 500, and Fig. 38b supports this conclusion, as the displacement pattern is quite different from that of a tank excited by the surge sway motion only. Further, the peak distribution shown in Fig. 38c indicates that the peaks are scattered irregularly and that the maximum peaks are concentrated at corners A and C. Based on the peak distribution and the free surface Fig. 38 Effect of heave motion on irregular waves, x x = x z = 2.3x 1, h = 5, x y = 2.0x 1. a The wave history of point A, b wave pattern (the figure in the top right corner is that for surge sway motion only), c distribution of peaks (the figure in the top right corner is that for surge sway motion only), d spectral analysis of point A at T = 0 400, e spectral analysis of point A at T =

27 48 J Mar Sci Technol (2011) 16:22 50 Fig. 39 The components of the forces of irregular waves (x x = x z = 2.3x 1 ) under coupled surge sway heave motion. h = 5 profiles, heave motion does not turn irregular waves into swirling waves. The dominant wave types are a combination of irregular and swirling-like waves during the whole simulation. Spectral analysis of point A, as shown in Figs. 38d, e, demonstrates that the heave motion not only reduces the strength of the third mode (x 3 = 2.13x 1 ) but it also increases the intensity of the first mode. The other two smaller modes, 1.3x 1 and 2.0x 1, are also observed after the heave motion starts to have an obvious effect (Fig. 38e), which indicates the complexity of the wave types generated under several blends of modes. Finally, the components of the forces are depicted in Fig. 39, and unlike the other types of sloshing waves, F x and F z seem to increase less rapidly when heave motion has an effect on the original irregular waves. 5 Conclusions The results obtained from an extensive simulation of the effects of excitation angles and heave motion on sloshing waves are reported in this paper. Based on these results, the following conclusions were reached: 1. Benchmark tests from previous studies indicate that the current numerical scheme has acceptable levels of accuracy. 2. The effects of the excitation angle on diagonal, single-directional, square-like and swirling waves have been demonstrated and discussed. The effect of excitation angle on square-like waves leads to differences between different areas of the terrace plane. The effect of excitation angle on irregular waves is not clear, since the odd natural modes of the irregular waves dominate the system, and they do not change with excitation angle. 3. The components of horizontal force for all types of sloshing waves under coupled surge sway motion are mainly dominated by sloshing displacement. The effects of excitation angle on the horizontal force components for both single-directional and squarelike waves are similar. For swirling waves, excitation angle has a significant influence on the beating periods and rotational directions of swirling waves, so the components of horizontal force for various excitation angles were shown to be dissimilar. 4. The classification of sloshing waves for various excitation frequencies, excitation angles and water depths was discussed in detail. The nonlinear effect of the sloshing displacement for a shallow water depth (d 0 /L = 0.15) causes more resonant modes, which in turn tend to trigger the generation of swirling waves. For d 0 /L = 0.5 and 1.0, the frequency domain of the swirling waves shrinks, but that of swirling-like waves extends. The coexistence of different types of sloshing was demonstrated by power spectral analyses for various water depths, particularly when multiple resonant modes affect the sloshing system. 5. The effect of coupling surge, sway and heave motions was also investigated in the present study, and the results show an unstable influence when the excitation frequency of the heave motion is twice as large as the fundamental natural frequency. 6. The sloshing wave pattern switches from singledirectional waves to counterclockwise-swirling waves once the heave motion begins to have an effect. For diagonal waves, heave motion significantly intensifies the sloshing displacement. The inclusion of heave motion triggers the occurrence of mode x 2,2, and may ultimately lead to the generation of swirling waves in the tank, even under diagonal forcing. 7. With heave motion, square-like waves ultimately turn into swirling waves, and unstable switching of the direction of swirl may lead to instability for moving vehicles or structures carrying tanks of fluid. 8. For swirling waves, heave motion significantly increases the sloshing displacement in the surge direction and diminishes the phenomenon where the direction of swirl flips. 9. Heave motion does not convert irregular waves into swirling waves, whereas swirling-like waves and irregular waves seem to coexist. 10. The components of the horizontal forces of various sloshing waves affected by heave motion are also presented. The results show that the influences of heave motion on the longitudinal (F x ) and transverse (F z ) forces depend on the types of sloshing waves and the excitation angle of the tank motion. The numerical method developed here can be extended to analyze the study of liquid sloshing in a tank with internal structures. Proper transformation functions are implemented to deal with numerical grids close to the internal structures. A study of liquid sloshing in a baffled