NONLINEAR MODELLING OF LIQUID SLOSHING IN A MOVING RECTANGULAR TANK

Transcription

1 NONLINEAR MODELLING OF LIQUID SLOSHING IN A MOVING RECTANGULAR TANK M. Serdar CELEBI * and Hakan AKYILDIZ! Abstract A nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated. The fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility. The tank is forced to move harmonically along a vertical curve with the rolling motion to simulate the actual tank excitation. The volume of fluid technique is used to track the free surface. The model solves the complete Navier-Stokes equations in primitive variables by use of the finite difference approximations. At each time step, a donar- acceptor method is used to transport the volume of fluid function and hence the locations of the free surface. In order to assess the accuracy of the method used, computations are verified through convergence tests and compared with the theoretical solutions and experimental results. Key Words: Sloshing, Free Surface Flow, Navier-Stokes Equations, Volume of Fluid Technique, Moving Rectangular Tank, Vertical Baffle, Finite Difference Method 1. INTRODUCTION Liquid sloshing in a moving container constitutes a broad class of problems of great practical importance with regard to the safety of transportation systems, such as tank trucks on highways, liquid tank cars on railroads, and sloshing of liquid cargo in oceangoing vessels. It is known that partially filled tanks are prone to violent sloshing under certain motions. The large liquid movement creates highly localized impact pressure on tank walls which may in turn cause structural damage and may even create sufficient moment to effect the stability of the vehicle which carries the container. When a tank is partially filled with fluid, a free surface is present. Then, rigid body acceleration of the tank produces a subsequent sloshing of the fluid. During this movement, it supplies energy to sustain the sloshing. There are two major problems arising in a computational approach to sloshing; these are the moving boundary conditions at the fluid tank interface, and the nonlinear motion of the free surface. * Assoc.Prof.Dr., Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering, 866, Maslak, Istanbul-TURKEY! Dr., Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering, 866, Maslak, Istanbul-TURKEY

2 Therefore, in order to include the nonlinearity and avoid the complex boundary conditions of moving walls, a moving coordinate system is used. The amplitude of the slosh, in general, depends on the nature, amplitude and frequency of the tank motion, liquid-fill depth, liquid properties and tank geometry. When the frequency of the tank motion is close to one of the natural frequencies of the tank fluid, large sloshing amplitudes can be expected. Sloshing is not a gentle phenomenon even at very small amplitude excitations. The fluid motion can become very non-linear, surface slopes can approach infinity and the fluid may encounter the tank top in an enclosed tanks. Hirt and Nichols (1981 developed a method known as the volume of fluid (VOF. This method allows steep and highly contorted free surfaces. The flexibility of this method suggests that it could be applied to the numerical simulation of sloshing and is therefore used as a base in this study. On the other hand, analytic study of the liquid motion in an accelerating container is not new. Abramson (1966 provides a rather comprehensive review and discussion of the analytic and experimental studies of liquid sloshing, which took place prior to The advent of high speed computers, the subsequent maturation of computational techniques for fluid dynamic problems and other limitations mentioned above have allowed a new, and powerful approach to sloshing; the numerical approach. Von Kerczek (1975, in a survey paper, discusses some very early numerical models of a type of sloshing problem, the Rayleigh-Taylor instability. Feng (1973 used a three-dimensional version of the marker and cell method (MAC to study sloshing in a rectangular tank. This method consumes large amount of computer memory and CPU time and the results reported indicate the presence of instability. Faltinsen (1974 suggests a nonlinear analytic method for simulating sloshing, which satisfies the nonlinear boundary condition at the free surface. Nakayama and Washizu (198 used a method basically allows large amplitude excitation in a moving reference frame. The nonlinear free surface boundary conditions are

3 addressed using an incremental procedure. This study employs a moving reference frame for the numerical simulation of sloshing. Sloshing is characterized by strong nonlinear fluid motion. If the interior of tank is smooth, the fluid viscosity plays a minor role. This makes possible the potential flow solution for the sloshing in a rigid tank. One approach is to solve the problem in the time domain with complete nonlinear free surface conditions (see Faltinsen Dillingham (1981 addressed the problem of trapped fluid on the deck of fishing vessels, which sloshes back, and fort and could result in destabilization of the fishing vessel. Lui and Lou (199 studied the dynamic coupling of a liquid-tank system under transient excitation analytically for a twodimensional rectangular rigid tank with no baffles. They showed that the discrepancy of responses in the two systems can obviously be observed when the ratio of the natural frequency of the fluid and the natural frequency of the tank are close to unity. Solaas and Faltinsen (1997 applied the Moiseev s procedure to derive a combined numerical and analytical method for sloshing in a general two-dimensional tank with vertical sides at the mean waterline. A low-order panel method based on Green s second identity is used as part of the solution. On the other hand, Celebi et al. (1998 applied a desingularized boundary integral equation method (DBIEM to model the wave formation in a three-dimensional numerical wave tank using the mixed Eulerian-Lagrangian (MEL technique. Kim and Celebi (1998 developed a technique in a tank to simulate the fully nonlinear interactions of waves with a body in the presence of internal secondary flow. A recent paper by Lee and Choi (1999 studied the sloshing in cargo tanks including hydro elastic effects. They described the fluid motion by higher-order boundary element method and the structural response by a thin plate theory. If the fluid assumed to be homogeneous and remain laminar, approximating the governing partial differential equations by difference equations would solve the sloshing 3

4 problem. The governing equations are the Navier-Stokes equations and they represent a mixed hyperbolic-elliptic set of nonlinear partial differential equations for an incompressible fluid. The location and transport of the free surface in the tank was addressed using a numerical technique known as the volume of fluid technique. The volume of fluid method is a powerful method based on a function whose value is unity at any point occupied by fluid and zero elsewhere. In the technique, the flow field was discretized into many small control volumes. The equations of motion were then satisfied in each control volume. At each time step, a donar-acceptor method is used to transport the fluid through the mesh. It is extremely simple method, requiring only one pass through the mesh and some simple tests to determine the orientation of fluid.. MATHEMATICAL MODELLING OF SLOSHING The fluid is assumed to be homogenous, isotropic, viscous and Newtonian and exhibits only limited compressibility. Tank and fluid motions are assumed to be two dimensional, which implies that there is no variation of fluid properties or flow parameters in one of the coordinate directions. The domain considered here is a rigid rectangular container with and without baffle configuration partially filled with liquid..1 Tank Motion Two different motion models that result in a two-dimensional tank excitation will be considered in the following subsections..1.1 Moving and Rolling Motion In this study, we will consider the tank excitation due to the motion of a tank moving with speed V(t along a vertical curve Y = ξ(x as shown in Figure 1. The following relations can be written: 4

5 U! (t = V(t! x V (t U! y(t = R V(t Ω(t = R (1 where the radius of curvature is, ξ R =, ( 3 / (1 + ξ XX X and the geometric constraint is, 1 / V(t = (1 + ξ X!. (3 x For a given tank speed V(t and vertical motion profile ξ(x, the horizontal movement of the tank X(t can be evaluated by solving Equation (3. Then, the basic modes of excitation y x + y U! x, U! ", and Ω where U = U î U ĵ can be obtained. To be more specifically, we shall assume a periodical motion profile for a study of harmonic excitation θo ξ ( X = coskx (4 k where θ o k is the elevation amplitude of the profile, k (=π/λ is the wave number of the profile and λ is the wavelength of the profile. The velocity is described by X! (t = V (1 + coskv t (5 o δθ o where V o is a characteristic tank speed and δ is a parameter of which characterized the response to the grade change. From Equations (1 (5, it can be shown that: o! 1 (t = V ωθ ( δsin sin (6 Ux o o U! (t = V ωθ cos (7 y o o Ω ( t = ωθo cos (8 5

6 where ω = (k V o is the characteristic frequency of excitation..1. Rolling Motion For a rolling motion about an axis on (X, Y = (, -d in Figure, U!,U!, and Ω are x y specified as: = dω! = dω Ω = θ! where θ is the angular displacement. It is assumed that U! U! x y (9 θ = θ o cos, where θ ο and ω represent the rolling amplitude and rolling frequency respectively.. General Formulation Before attempting to describe the governing equations, it is necessary to impose the appropriate physical conditions on the boundaries of the fluid domain. On the solid boundary, the fluid velocity equals the velocity of the body. u " u = V =, = V " t n t n = (1 where V " n and V " t are normal and tangential components and u and v are the horizontal (x and vertical (y components of the fluid velocity respectively. The location of the free surface is not known priori and presents a problem when the boundary conditions are to be applied. If the free surface boundary conditions are not applied at the proper location, the momentum may not be conserved and this would yield incorrect results. Tangential stresses are negligible because of the larger fluid density comparing with the air. The only stress at such a surface is the normal pressure. Therefore, the summation of the forces normal to the free surface must be balanced by the atmospheric pressure. This yields the dynamic boundary condition at a free surface 6

7 ( ρu ( ρu ( ρv ( ρv P = PATM + ν n xmx + n xmy + + n ymy = (11 x y x x where P ATM, ν and ρ are atmospheric pressure, kinematic viscosity and density of the fluid and n x, m x are the horizontal components of the unit vector, normal and tangent to the surface respectively and similarly, n y, m y are the vertical components of the unit vector, normal and tangent to the surface respectively. In addition, it is necessary to impose the kinematic boundary condition that the normal velocity of the fluid and the free surface are equal. In this study, unsteady motion takes place and the characteristic time during the flow changes may be very small. Therefore, compressibility of fluid may not be ignored. In some cases, it is desirable to assume that the pressure is a function of density. dp = dρ c (1 where c is the adiabatic speed of sound. Expanding the mass equation about the constant mean density ρ ο and retaining only the lowest order terms, yields 1 c ρ " + ρ V t =. (13 The forces acting on the fluid in order to conserve momentum must balance the rate of change of momentum of fluid per unit volume. This principle is expressed as t " " " ( ρv + V ( ρv = p + F + ν ( ρv, " " (14 where p is the pressure and F " is the body force(s acting on the fluid..3 The Coordinate System and Body Forces In order to include the non-linearity and avoid the complex boundary conditions of moving walls, the moving coordinate system is used. The origin of the coordinate system is in the position of the center plane of the tank and in the undisturbed free surface. The moving coordinate is translating and rotating relative to an inertial system (see Figure. The 7

8 equilibrium position of the tank relative to the axis of rotation is defined by γ. For instance, the tank is rotating about a fixed point on the y-axis at γ = 9 o. Thus the moving coordinate system can be used to represent general roll (displayed by θ or pitch of the tank. We suppose that the moving frame of reference is instantaneously rotating with an " angular velocity Ω ( θ! about a point O which itself is moving relative to the Newtonian frame " with the acceleration U!. The absolute acceleration of an element is then, " A "! " * = U + a (15 where where * a " is the acceleration of an element relative to the point O. Here " r t * a " is represented by " " " r " r Ω " " " = + Ω + r + Ω ( Ω r (16 t t t "* " a " = a is the acceleration of an element relative to the translating and rotating frame " of reference and r = u " * t acceleration of an element is thus, is the velocity of the element in this frame. The absolute " " " " " = U! + a + θ! u +!! θ r + θ! (17 A * " ( θ! r This expression may be equated to the local force acting per unit mass of fluid to give " the equation of motion in the moving frame. Here, U! is simply the apparent body force such as drift force;! u " θ is the deflecting or coriolis force;! " θ r is referred to as the Euler force and! " ( θ! r θ is the centrifugal force. Thus, the body force term in the Equation (14 is expressed in component form as F x F y = gsin θ! θ y + θ! x d(!! θsin γ θ! cos γ U! θ! v (18a = gcosθ +! θ x + θ! y + d(!! θcos γ + θ! sin γ U! + θ! u (18b x y 8

9 where g is the gravitational acceleration, d is the distance between the origin of the moving coordinate and the axes of rotation, y directions given in Equations (1 and (9. U! x and U! y are the accelerations of the tank in the x and The governing Equation (13 of the fluid motion with the limited compressibility option (Equation 1 yields to following equation by normalizing the fluid mean density to one 1 c p u v + + t x y = (19 where all variables are now defined in the tank-fixed coordinate system. The modified momentum equations yield the required expressions for two-dimensional flow in a rotating tank u u u p + u + v + = gsin θ θ! v!! θ( y + dsin γ U! ( x dcos ( x + θ! + γ + ν u, (a t x y x v v v p + u + v + = gcosθ + θ! u +!! θ! y. (b t x y y ( x + dcos γ U! + θ ( y + dsin γ + ν( v 3. NUMERICAL STABILITY AND ACCURACY In this section the strengths and weaknesses of the numerical technique that effect the stability and accuracy as well as the limitation on the extent of computation will be discussed. In the numerical study, the flow field is discretized into many small control volumes. The equations of motion are then satisfied in each small control volume. Obvious requirements for the accuracy are included the necessity for the control volumes or cells to be small enough to resolve the features of interest and for time steps to be small enough to prevent instability. Once a mesh has been chosen, the choice of the time increment necessary for the numerical stability is governed by two restrictions: One of them is that the fluid particles can not move through more than one cell in one time step, because the difference equations are assumed the 9

10 fluxes only between adjacent cells. Therefore, the time increment must satisfy the following inequality, δx i+ δyi t < min 1 /, 1 / (1 Ui, j Vi, j δ + where δ i 1 / and δ i 1 / are the half sizes of the cell in x and y directions respectively. x + y + Typically δ t is chosen equal to a time between one-fourth and one-third of the minimum cell transit time. The second restriction is that, for a non-zero value of kinematic viscosity, momentum must not diffuse more than approximately one cell in one time step. A linear stability analysis shows that this limitation implies xi i j y j 1 δ δy νδ t < ( δx + δ The other parameter necessary to insure numerical stability is α, which is the upstream differencing parameter. In the absence of physical viscosity, α must be included for the stability. It can also be seen how α can be adjusted to minimize diffusion-like truncation error. The proper choice for α is then, U i, jδt Vi, jδt 1 α max, (3 δx i+ 1/ δy i+ 1/ For our computations, α is typically set to be 3-5% higher than the Courant number. Formally, when considering accuracy of a finite difference scheme, the order of accuracy is defined by the lowest order powers of the increments of time and space appearing in the truncation error of the modified equation. A higher order scheme, which is second order accurate, can be used to improve accuracy, but any process, which increases the accuracy of the results, will also increase the computation time, and in most cases the relationship is nonlinear. Another parameter, which has an effect on the accuracy, is σ. This is the criterion used to govern the level of mass conservation. For an incompressible fluid, 1

11 u " σ (4 If σ is not zero, then the fluid is numerically compressible. Since it is extremely difficult to enforce zero divergence, a finite value must be used. Typical values are about σ = 1 3. But it has been found that even larger values of epsilon do not seriously affect the results. 4. THEORETICAL ANALYSIS The procedure of theoretical solutions used in comparison to the numerical results is briefly summarized in this section. For a rectangular tank without any internal obstacles under combined external excitations (e.g. sway plus roll or surge and pitch, analytical solutions can be derived from the fundamental governing equations of fluid mechanics. These solutions can be used to predict liquid motions inside the tank, the resultant dynamic pressures on tank walls, and the effect of phase relationship between the excitations on sloshing loads. The case is considered as a two-dimensional, rigid, rectangular tank without internal obstacles that is filled with inviscid, incompressible liquid. It is forced to oscillate harmonically with a horizontal velocity U x, vertical velocity U y, and a rotational velocity Ω. Since the fluid is incompressible, the velocity potential must satisfy the Laplace equation with the boundary conditions on the tank walls. The dynamic and kinematic free surface boundary conditions must, then, be satisfied on the instantaneous free surface. For the analytical solutions, it is assumed that: (i the amplitudes of tank motion Ω, U x and U y are of the same order of magnitudes, being proportional to a small parameter ε << 1, (ii Ω, U x and U y oscillate at the same frequency but in different phases. Once the velocity potential φ has been determined, the pressure of the fluid can be calculated from the Bernoulli s equation. 11

12 5. NUMERICAL IMPLEMENTATIONS It is assumed that the mesh dimensions would be small enough to resolve the main feature of liquid sloshing in each case. The step of time advance t, in each cycle is also assumed to be so small that no significant flow change would occur during t. There is no case where a steady state solution is reached in the forcing periods used. Either instability set in or computer time becomes excessive, so the duration of computation is limited for each case. Therefore, computations are halted when the fluid particles extremely interact and spray over the top side of the tank during the extreme sloshing. In all cases the tank starts to roll about the centre of the tank bottom at time t = +. Since the major concern is to find the peak wave elevations on the left wall of the tank on the free surface, and forces and moments on both walls, the analysis is based on the comparison of the wave elevations above the calm free surface and corresponding forces and moments exerted. 5.1 Moving Rectangular Tank For the numerical solutions with the moving rectangular tank along a vertical curve, D the β value (β = a of.65 is used. For fill depth D = 4 ft, the effective tank length (a corresponding to β value is 3 ft.. The parameter δ (given in Equation 5 is chosen to equal to 1 showing the variation of tank speed with acceleration downhill and declaration uphill. The tank speed V o (given in Equation 5 is taken as 7.5 ft/sec.. In the following numerical computations, the excitation frequency of the tank ω is varied from.1 to 1.3 rad/sec. and the corresponding wavelength of the periodical motion profile is taken as 1.43 ft.. Tank roll motion is defined by θ = θ o sin where θ ο is the rolling amplitude. A typical numerical simulation of sloshing in a rigid rectangular tank with and without baffle is shown in Figure 3 for the case of resonant frequency of the fluid, ω n =

13 rad/sec., and rolling amplitude θ ο = 8 ο. Αs can be seen from the snapshots, the baffled case decreased the amplitude of sloshing but generated some additional eddies near the baffle. Figure 4 through 8 show the plots of the wave amplitude, horizontal and vertical acceleration of the tank, sloshing forces and moments in the longitudinal directions in connection with the rolling of the tank. For the lower frequency of the tank excitation, ω =.1 rad/sec. and the rolling amplitudes θ ο = 5.73 ο and θ ο = 8 ο, in Figure 4, the normalized sloshing force and moment are only slightly dependent on the amplitude of excitation. For the sloshing force without baffle (see in Figure 4c, the numerical and theoretical results agreed well at the rolling amplitude θ ο = 5.73 ο. The wave profile exhibits linear characteristics due to the lower excitation frequency. During the simulation, computations show that the numerical result of maximum force (F max gives %7 overestimate compared with the theoretical solution. It can be concluded from the baffled results that, for the lower excitation frequency, sloshing force and moment reduced slightly compared to those of unbaffled case. As the excitation frequency of ω is increased to.5, results start to diverge from linear characteristics as shown in Figure 5. The normalized sloshing force and moment are deviated slightly for the rolling amplitudes θ ο = 5.73 ο and θ ο = 8 ο due to the still existing linear effects. Results show that there is a significant difference between theoretical (linear solution and nonlinear numerical force calculations especially near the = 1.5 ~.5 due to the dominant hydrostatic effect corresponding to the wave amplitude. It is also seen that there is a %36 increase in maximum forces (F max between rolling frequencies ω =.1 and.5 rad/sec.. In the case of baffle configuration (in Figures 5g-h, the maximum forces and moments are reduced %19 and %4 compared to the unbaffled cases respectively. We also observed that the effect of baffle on sloshing force and moment increased as the rolling frequency changes from ω =.1 to.5 rad/sec.. 13

14 It is seen, in Figure 6, that the normalized sloshing force and moment are increased depending on the amplitude of excitation due to the non-linear effects as the parameter range ( increases in connection with the tank motion along a vertical curve. In baffled case (in Figures 6g-h, the maximum forces and moments are reduced %114 and %7 (for the rolling amplitude θ ο = 5.73 ο compared to the unbaffled cases respectively. It can be concluded from the results that the maximum effect of baffle occurs near the resonance frequency of the fluid (ω n = rad/sec.. On the resonance frequency, in Figure 7, the magnitude of the maximum forces and moments did not change significantly compared with the rolling frequency ω =.9 rad/sec., but the effect of baffle becomes less due to the increasing sloshing effects (turbulent eddies, wave breaking and spraying near the baffle (see in Figures 7g-h. On the off-resonant frequency ω = 1.3 rad/sec., in Figure 8, it is observed that the sloshing effects are significantly reduced (% 86 in force and %97 in moment for the rolling amplitude θ ο = 5.73 ο compared with the resonant frequency near the baffle. For the larger rolling amplitude (θ ο = 8 ο, new values become %4 in force and %37 in moment. One possible reason may be that the increasing effect of turbulence is reduced the baffle effect due the increased pressure gradient variations on the baffle surface. In order to show the impact of the vertical baffle located at the mid-bottom of the rectangular rigid tank, the percentage of reduction for forces and moments between unbaffled and baffled cases was computed for different rolling amplitudes as shown in Figures 9 and 1. The vertical axes which is represented by % F and % M calculated as F unbaffled baffled F =, F F unbaffled M = M unbaffled M M unbaffled baffled. It can be noted from the Figures 9 and 1 that the amount of reduction in force and moment is increased as the rolling frequency ω approaches to the 14

15 resonant frequency of the fluid where ω n is rad/sec.. It can be also observed that the reduction in force and moment is started to decrease after the rolling frequency passed the resonant frequency. 5. Rolling Rectangular Tank For the numerical solutions with the rigid tank in roll motion, the β values D (β = a of.65 and 1. are used. The first value of β corresponds to shallow water case, and the second is of a typical intermediate fill depth. The frequency of roll excitation (ω and amplitude (θ o are varied from.6 to 1. rad/sec. and 4 o to 8 o respectively. For a tank width of 6 ft., fill depth gets values of 7.5 ft. and 3 ft. in terms of β value. The parameter d, which defines the location of the center of roll motion of the rigid tank is chosen to be equal to 1. and. ft. The resonance frequency of the fluid inside the tank is ω n = 1.43 rad/sec.. Starting with a tank of intermediate fill depth (β = 1 and with the rolling axes located at the bottom of the tank (d = 1, a typical snapshot for the simulation of the sloshing with and without baffle configuration is given in Figure 11. Figures 1 through 15 show the plots of wave profiles, angular accelerations, sloshing force and turning moment in a rigid tank due to the roll motion. For the lower frequency ω =.6 rad/sec., as shown in Figures 1c and 1d, the normalized sloshing forces and moments for the unbaffled case are only very slightly dependent on the amplitude of excitations θ ο = 4 ο and 8 ο. It is seen that there is a significant difference between theoretical and numerical results in Figures 1c and d due to the nonlinear and viscous effects in numerical model and perturbation technique used in the theoretical solutions. In the unbaffled case, in Figures 1g and h, the sloshing effects become 15

16 slightly dominate in terms of increasing rolling amplitude θ ο. It can also be concluded that the correct arrangement of baffle is reduced the sloshing force and moment %.7 and %11 respectively for θ ο = 4 ο, and %3. and %7.3 for θ ο = 8 ο. As ω is increased from.6 to 1. rad/sec., which is closer to the resonant frequency of 1.43 rad/sec., theoretical results in Figures 13c and d show non-linear behaviour. The nonlinear behaviour is more pronounced for larger θ ο. Numerical solutions indicate that the trough of wave profile gets relatively wider and flat shape with the increasing of rolling amplitude θ ο and rolling frequency ω. Figures 13c and d reveal that there are two basic differences between theoretical and numerical results: first, the phase shifting is more pronounced, and second, asymmetric behaviour starts to dominate. The severe oscillations of fluid particles especially around the baffle and right wall of the tank occur with the increasing rolling amplitude and frequency as shown in Figures 13g and h. In Figure 14, for the lower fill depth case (β =.65 and D = 7.5 ft., the non-linear effects such as narrow zero crossing and larger amplitudes in force and moment and increased phase shifting are observed in theoretical and numerical results compared to the previous β = 1. case. As a result of this, forces and moments in both numerical and theoretical solutions are obtained larger (for instance; %3. and %144 are increased for the case of theoretical computation of force and moment and the locations of maximum force and moment are also shifted for the rolling amplitude θ ο = 4 ο. These effects can be observed in Figures 14c and d more severely as the rolling amplitude increased to θ ο = 8 ο. It must be noted that the effect of vertical baffle for the lower fill depth case is greatly reduce the over turning moment (for instance; %56 decreased between baffled and unbaffled cases for β = 1 and θ ο = 4 ο, %137 decreased between baffled and unbaffled cases for β =.65 and θ ο = 4 ο and sloshing effects (see Figures 13-14g and h. 16

17 The location of rolling center characterized by the parameter d plays an important role on the magnitude of the sloshing effects as shown in Figure 15. In order to show the effect of d variation, the typical case of β = 1 and ω = 1. rad/sec. is selected. The variation of θ ο from 4 ο to 8 o is increased the numerical over turning moment by %5.3. The numerical results indicate that the overall magnitudes of sloshing forces and moments are hardened with the increasing of d. In order to observe the effect of roll frequency on the maximum wave height and compare the theoretical and experimental results, the case of D/a =.5 and d/a =.5 with the rolling amplitudes 6 o and 8 o is selected, as shown in Figure 16. It can be noted that, with the increasing off-resonance rolling frequency, the theoretical, experimental and numerical results tend to agree well. On the other hand, around the resonance frequency, the normalized wave elevation is obtained relatively low than those of experimental and theoretical results. One possible reason for this may be the effect of viscosity in the numerical model used. It is known that the theoretical model did not contain the viscous effects and the experimental model did not match the Reynolds number. Additional comparisons are presented in Figure 17 for a tank length of ft. (a = ft. at a water depth of.8 ft.. The analytical, experimental and numerical solutions indicate that the normalized wave amplitude is linearly proportional to the excitation rolling amplitude. The agreement between numerical solutions and experimental results is very well comparing to the analytical solutions. 6. CONCLUSIONS The volume of fluid technique has been used to simulate two-dimensional viscous liquid sloshing in moving rectangular baffled and unbaffled tanks. The VOF method was also used to track the actual positions of the fluid particles on the complicated free surface. The liquid was assumed to be homogeneous and to remain laminar. The excitation was assumed 17

18 harmonic, after the motion was started from the rest. A moving coordinate system fixed in the tank was used to simplify the boundary condition on the fluid tank interface during the large tank motions. The general features of the effects of baffles on liquid sloshing inside the various partially filled tanks were studied. Analytical solutions for liquid sloshing under combined excitations were compared with both numerical and some experimental results. The following conclusions can be drawn from our numerical computations: i The liquid is responded violently causing the numerical solution to become unstable when the amplitude of excitation increased. The instability may be related to the fluid motion such as the occurrence of turbulence, the transition from homogeneous flow to a two-phase flow and the introduction of secondary flow along the third dimension. Thus, the applicability of the method used in the present study is limited to the period prior to the inception of these flow perturbations. ii The liquid sloshing inside a tank revealed that flow over a vertical baffle produced a shear layer and energy was dissipated by viscous action. iii The effect of vertical baffles was most pronounced in shallow water. For this reason, especially the over turning moment was greatly reduced. iv The increased fill depth, the rolling amplitude and frequency of the tank with/without baffle configuration directly effected the degree of non-linearity of the sloshing phenomena. As a result of this, the phase shifting in forces and moments occurred. v The larger forces and moments were obtained with the reducing fill depth due to the increasing free surface effect. Finally, the effects of turbulence and two phase flow (sprays, drops and bubbles in the post impact period as well as three dimensional effects need to be incorporated to assure a stable and reliable modeling for such cases. For future work, second-order representation of 18

19 derivatives may be employed to better approximate to the rapid change of divergence in the fluid. The effect of speed of sound, on the case of extreme sloshing, has to be checked to see the compressibility effect in some degree. Model studies for sloshing under multi-component random excitations with phase difference should be carried out to investigate sloshing loads under more realistic tank motion inputs. Additionally, an integrated design synthesis technique must be developed to accurately predict sloshing loads for design applications. V " V " n t NOMENCLATURE : The normal component of the fluid velocity : The tangential component of the fluid velocity P : Fluid pressure P : Atmospheric pressure ATM ν : Kinematic viscosity ρ : Fluid density n x, m x : The horizontal components of the unit vector, normal and tangent to the surface n y, m y : The vertical components of the unit vector, normal and tangent to the surface F " : Body forces θ : Roll angle θ : Roll amplitude of the tank o γ : The equilibrium angle of the tank relative to the axis of rotation d : The distance between the origin of the moving coordinate and the axis of rotation D : Fill depth a : Tank length Ω " : Angular velocity "! Acceleration of the moving frame U : * a " δt : Acceleration of an element relative to the point O : Time increment α : The upstream differencing parameter ε : The perturbation parameter 19

20 σ : The compressibility parameter ω : Roll frequency of the tank ωn : Natural frequency of the fluid inside the tank φ : The velocity potential of the fluid δ: The response parameter of the grade change V o : The characteristic tank speed k: The wave number of the motion profile X(t: The horizontal movement of the tank V(t: The speed of the moving tank along the motion profile U! U! x y : The tank acceleration in x direction : The tank acceleration in y direction R: The radius of curvature of the motion profile ξ (X : The vertical motion profile η: The wave amplitude ACKNOWLEDGEMENT The authors would like to thank to Research Fund of Istanbul Technical University for the financial support of this study. REFERENCES Abramson, H.N., Dynamic Behavior of Liquids in Moving Containers with Application to Space Vehicle Technology. NASA-SP-16. Celebi, M.S., Kim, M.H., Beck, R.F., Fully Non-linear 3-D Numerical Wave Tank Simulation. J. of Ship Research, Vol.4, No.1, pp

21 Kim, M.H., Celebi, M.S., Kim, D.J., Fully Non-linear Interactions of Waves With a Three-Dimensional Body in Uniform Currents. Applied Ocean Research, Vol., pp Dillingham, J., Motion Studies of a Vessel with Water on Deck. Marine Technology, SNAME, Vol.18, No.1, pp Faltinsen, O.M., A Non-linear Theory of Sloshing in Rectangular Tanks. J. of Ship Research, Vol.18, No.4, pp Faltinsen, O.M., A Numerical Non-linear Method of Sloshing in Tanks With Two- Dimensional Flow. J. of Ship Research, Vol., No.3, pp Feng, G.C., Dynamic Loads Due to Moving Liquid. AIAA Paper No: Hirt, C.W., Nichols, B.D., Volume of Fluid Method for the Dynamics of Free Boundaries. Journal of Computational Physics, Vol.39, pp Lee, D.Y., Choi, H.S., Study on Sloshing in Cargo Tanks Including Hydro elastic Effects. J. of Mar. Sci. Technology, Vol.4, No.1. Lou, Y.K., Su, T.C., Flipse, J.E., 198. A Non-linear Analysis of Liquid Sloshing in Rigid Containers. US Department of Commerce, Final Report, MA-79-SAC-B18. Lui, A.P., Lou, J.Y.K., 199. Dynamic Coupling of a Liquid Tank System Under Transient Excitations. Ocean Engineering, Vol.17, No.3, pp Nakayama, T., Washizu K., 198. Nonlinear Analysis of Liquid Motion in a Container Subjected to Forced Pitching Oscillation. Int. J. for Num. Meth. in Eng., Vol.15, pp Solaas, F., Faltinsen, O.M., Combined Numerical and Analytical Solution for Sloshing in Two-Dimensional Tanks of General Shape. Vol.41, No., pp Von Kerczek, C.H., Numerical Solution of Naval Free-Surface Hydrodynamics Problems. 1 st International Conference on Numerical Ship Hydrodynamics, Gaithersburg, USA. 1

22 List of Figures Figure 1. The motion profile of the tank Figure. The moving coordinate system Figure 3. A snapshot for numerical simulation of the shallow water sloshing (ω =1.864 rad/sec., θ o = 8 o Figure 4. The comparison of the unbaffled and baffled cases with the various parameters (ω =.1 rad/sec., β =.65, d = 1 ft. Figure 5. The comparison of the unbaffled and baffled cases with the various parameters (ω =.5 rad/sec., β =.65, d = 1 ft. Figure 6. The comparison of the unbaffled and baffled cases with the various parameters (ω =.9 rad/sec., β =.65, d = 1 ft. Figure 7. The comparison of the unbaffled and baffled cases with the various parameters (ω = rad/sec., β =.65, d = 1 ft. Figure 8. The comparison of the unbaffled and baffled cases with the various parameters (ω = 1.3 rad/sec., β =.65, d = 1 ft. Figure 9. The effect of the vertical baffle on forces and moments (θ o = 5.73 o Figure 1. The effect of the vertical baffle on forces and moments (θ o = 8 o Figure 11. A snapshot for numerical simulation of the intermediate fill depth sloshing (ω =1. rad/sec., θ o = 8 o Figure 1. The comparison of the unbaffled and baffled cases with the various parameters (ω =.6 rad/sec., β = 1, d = 1 ft. Figure 13. The comparison of the unbaffled and baffled cases with the various parameters (ω = 1. rad/sec., β = 1, d = 1 ft.

23 Figure 14. The comparison of the unbaffled and baffled cases with the various parameters (ω = 1. rad/sec., β =.65, d = 1 ft. Figure 15. The comparison of the unbaffled and baffled cases with the various parameters (ω = 1. rad/sec., β = 1, d = ft. Figure 16. The effect of roll frequency on maximum wave height Figure 17. Comparisons of analytical solutions, experimental results and numerical solutions. Maximum wave amplitude vs roll angle. 3

24 Y y, U y x, U x D a V(t Motion Profile Figure 1. The Motion Profile of the Tank. x -y : Equilibrium Position x - y : Instantaneous Position x X y y Y θ γ d a D x O X Figure. The Moving Coordinate System 4

25 Time =.59 sec. Time =.59 sec. (a (e Time =. sec. Time =. sec. (b (f Time = 4.1 sec. Time = 4.1 sec. (c (g Time = 5.5 sec. Time = 5.5 sec. (d Figure 3. A Snapshot for Numerical Simulation of the Shallow Water Sloshing. ( ω=1.864 rad/sec, θ (h 5

26 (a Wave Profile ( θ =5.73 Wave Profile ( θ (e Wave Profile-Baffled ( θ =5.73 Wave Profile-Baffled ( θ ζ /a -.36 ζ /a Acceleration (b Horizontal ( θ =5.73 Vertical ( θ =5.73 Horizontal ( θ Vertical ( θ (c Force ( θ =5.73 Force ( θ Force-Theoretical ( θ =5.73 Acceleration (f Horizontal-Baffled ( θ =5.73 Vertical-Baffled ( θ =5.73 Horizontal-Baffled ( θ Vertical-Baffled ( θ (g Force-Baffled ( θ =5.73 Force-Baffled ( θ F/(.5ρ gd θ F max1 =16.8 F max =16.81 F max3 =15.68 F/(.5ρ gd θ F max1 =15.84 F max = (d Moment ( θ =5.73 Moment ( θ 15 1 (h Moment-Baffled ( θ =5.73 Moment-Baffled ( θ M/(.5ρ gd 3 θ M max1 =8.5 M max =8.74 M/(.5ρ gd 3 θ M max1 =7.7 M max = Figure 4. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω =.1rad/sec,β =.65,d=1ft 6

27 (a Wave Profile ( θ =5.73 Wave Profile ( θ (e Wave Profile-Baffled ( θ =5.73 Wave Profile-Baffled ( θ ζ /a -.36 ζ /a Acceleration (b Horizontal ( θ =5.73 Vertical ( θ =5.73 Horizontal ( θ Vertical ( θ (c Force ( θ =5.73 Force ( θ Force-Theoretical ( θ =5.73 Acceleration (f Horizontal-Baffled ( θ =5.73 Vertical-Baffled ( θ =5.73 Horizontal-Baffled ( θ Vertical-Baffled ( θ (g Force-Baffled ( θ =5.73 Force-Baffled ( θ F/(.5ρ gd θ F max1 =3. F max3 =.51 F max =13.67 F/(.5ρ gd θ F max1 =19.8 F max = (d Moment ( θ =5.73 Moment ( θ (h Moment-Baffled ( θ =5.73 Moment-Baffled ( θ M/(.5ρ gd 3 θ M max1 =1.6 M max =1.78 M/(.5ρ gd 3 θ M max1 =9.7 M max = Figure 5. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω =.5 rad/sec, β =.65, d = 1 ft 7

28 (a Wave Profile ( θ =5.73 Wave Profile ( θ (e Wave Profile-Baffled ( θ =5.73 Wave Profile-Baffled ( θ ζ /a -.36 ζ /a (b Horizontal ( θ =5.73 Vertical ( θ =5.73 Horizontal ( θ Vertical ( θ (f Horizontal-Baffled ( θ =5.73 Vertical-Baffled ( θ =5.73 Horizontal-Baffled ( θ Vertical-Baffled ( θ.8.8 Acceleration Acceleration (c Force ( θ =5.73 Force ( θ 5 4 (g Force-Baffled ( θ =5.73 Force-Baffled ( θ 3 3 F/(.5ρ gd θ F max1 =37.13 F max =46.5 F/(.5ρ gd θ F max1 =17.8 F max = (d Moment ( θ =5.73 Moment ( θ (h Moment-Baffled ( θ =5.73 Moment-Baffled ( θ M/(.5ρ gd 3 θ M max1 =5.86 M max =43.6 M/(.5ρ gd 3 θ M max1 =8.41 M max = Figure 6. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω =.9rad/sec,β =.65, d = 1 ft 8

29 (a Wave Profile ( θ =5.73 Wave Profile ( θ (e W. Profile-Baffled ( θ =5.73 W. Profile-Baffled ( θ ζ /a -.36 ζ /a Acceleration (b Horizontal ( θ =5.73 Vertical ( θ =5.73 Horizontal ( θ Vertical ( θ (c Force ( θ =5.73 Force ( θ Acceleration (f Horizontal-Baffled ( θ =5.73 Vertical-Baffled ( θ =5.73 Horizontal-Baffled ( θ Vertical-Baffled ( θ (g Force-Baffled ( θ =5.73 Force-Baffled ( θ F/(.5ρ gd θ F/(.5ρ gd θ F max1 =37.6 F max = F max1 =9.3 F max = (d Moment ( θ =5.73 Moment ( θ 5 15 (h Moment-Baffled ( θ =5.73 Moment-Baffled ( θ M/(.5ρ gd 3 θ M max1 =7.37 M max =33.68 M/(.5ρ gd 3 θ M max1 =16.4 M max = Figure 7. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω = rad/sec, β =.65, d = 1 ft 9

30 (a Wave Profile ( θ =5.73 Wave Profile ( θ (e Wave Profile-Baffled ( θ =5.73 Wave Profile-Baffled ( θ ζ /a -.36 ζ /a Acceleration (b Horizontal ( θ =5.73 Vertical ( θ =5.73 Horizontal ( θ Vertical ( θ Acceleration (f Horizontal-Baffled ( θ =5.73 Vertical-Baffled ( θ =5.73 Horizontal-Baffled ( θ Vertical-Baffled ( θ (c Force ( θ =5.73 Force ( θ (g Force-Baffled ( θ =5.73 Force-Baffled ( θ F/(.5ρ gd θ F/(.5ρ gd θ F max1 =3.76 F max = F max1 =15.79 F max = (d Moment ( θ =5.73 Moment ( θ 5 15 (h Moment-Baffled ( θ =5.73 Moment-Baffled ( θ M/(.5ρ gd 3 θ M max1 =1.5 M max =3.96 M/(.5ρ gd 3 θ M max1 =8.1 M max = Figure 8. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω = 1.3 rad/sec, β =.65, d = 1 ft 3

31 Reduction of Sloshing Force Reduction of Sloshing Moment Resonant Frequency ( ω n =1.864 % F, % M Tank Length, a = 3 ft D / a =.15 d / a = Rolling Frequency, ω Figure 9. The Effect of the Vertical Baffle on Forces and Moments. ( θ = Reduction of Sloshing Force Reduction of Sloshing Moment Resonant Frequency ( ω n =1.864 % F, % M Tank Length, a=3 ft D / a =.15 d / a = Rolling Frequency, ω Figure 1. The Effect of the Vertical Baffle on Forces and Moments. ( θ 31

32 Time =.758 sec. Time =.758 sec. (a Time =.51 sec. (e Time =.51 sec. (b Time = 4.5 sec. (f Time = 4.5 sec. (c Time = 6.1 sec. (g Time = 6.1 sec. (d (h Figure 11. A Snapshot for Numerical Simulation of the Intermediate Fill Depth Sloshing ( ω=1. rad/sec, θ 3

33 (a Wave Profile ( θ =4 Wave Profile ( θ (e Wave Profile-Baffled ( θ =4 W. Profile-Baffled ( θ ζ /a -.4 ζ /a Acceleration (b Horizontal ( θ =4 Vertical ( θ =4 Horizontal ( θ Vertical ( θ Acceleration (f Horizontal-Baffled ( θ =4 Vertical-Baffled ( θ =4 Horizontal-Baffled ( θ Vertical-Baffled ( θ F/(.5ρ gd θ M/(.5ρ gd 3 θ F max1 =7.9 F max =7.69 F max3 =5.45 F max4 =5.46 M max1 =4.1 M max =4.14 M max3 =.55 M max4 =.56 (c Force ( θ =4 Force ( θ Force-Theoretical ( θ =4 Force-Theoretical ( θ (d Moment ( θ =4 Moment ( θ Moment-Theoretical ( θ =4 Moment-Theoretical ( θ F/(.5ρ gd θ M/(.5ρ gd 3 θ F max1 =7.65 F max =7.45 (g Force-Baffled ( θ =4 Force-Baffled ( θ M max1 =3.68 M max =3.86 (h Moment-Baffled ( θ =4 Moment-Baffled ( θ = Figure 1. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω =.6 rad/sec, β =1,d=1ft 33

34 .5 (a.15 (e..15 Wave Profile ( θ =4 Wave Profile ( θ.1.9 Wave Profile-Baffled ( θ =4 Wave Profile-Baffled ( θ ζ /a ζ /a Acceleration (b Horizontal ( θ =4 Vertical ( θ =4 Horizontal ( θ Vertical ( θ Acceleration (f Horizontal-Baffled ( θ =4 Vertical-Baffled ( θ =4 Horizontal-Baffled ( θ Vertical-Baffled ( θ F/(.5ρ gd θ M/(.5ρ gd 3 θ F max1 =9.58 F max =9.4 F max3 =14.43 F max4 =15.7 M max1 =4.94 M max =5.1 M max3 =4.76 M max4 =5.39 (c Force ( θ =4 Force ( θ Force-Theoretical ( θ =4 Force-Theoretical ( θ (d Moment ( θ =4 Moment ( θ Moment-Theoretical ( θ =4 Moment-Theoretical ( θ F/(.5ρ gd θ M/(.5ρ gd 3 θ F max1 =6. F max =7.86 (g Force-Baffled ( θ =4 Force-Baffled ( θ M max1 =3.17 M max =4. (h Moment-Baffled ( θ =4 Moment-Baffled ( θ Figure 13. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω =1.rad/sec,β =1, d=1ft 34

35 .8 (a.8 (e.6 Wave Profile ( θ =4 Wave Profile ( θ.6 Wave Profile-Baffled ( θ =4 Wave Profile-Baffled ( θ ζ /a ζ /a (b (h Horizontal ( θ =4 Vertical ( θ =4 Horizontal ( θ Vertical ( θ (f Horizontal-Baffled ( θ =4 Vertical-Baffled ( θ =4 Horizontal-Baffled ( θ Vertical-Baffled ( θ Acceleration Acceleration F/(.5ρ gd θ M/(.5ρ gd 3 θ F max3 =14.9 F max4 =17.7 F max1 =.5 F max =4.6 M max3 =1.1 M max4 =13.4 M max1 =1.84 M max =16.46 (c Force-Theoretical ( θ =4 Force-Theoretical ( θ Force ( θ =4 Force ( θ (d Moment-Theoretical ( θ =4 Moment-Theoretical ( θ Moment ( θ =4 Moment ( θ F/(.5ρ gd θ M/(.5ρ gd 3 θ F max1 =11.1 F max =1.71 (g Force-Baffled ( θ =4 Force-Baffled ( θ M max1 =5.11 M max =6.16 (h Moment-Baffled ( θ =4 Moment-Baffled ( θ Figure 14. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω = 1. rad/sec, β =.65, d = 1 ft 35

36 (a Wave Profile ( θ =4 Wave Profile ( θ (e Wave Profile-Baffled ( θ =4 Wave Profile-Baffled ( θ ζ /a -.5 ζ /a Acceleration (b Horizontal ( θ =4 Vertical ( θ =4 Horizontal ( θ Vertical ( θ (c Force ( θ =4 Force ( θ Acceleration (f Horizontal-Baffled ( θ =4 Vertical-Baffled ( θ =4 Horizontal-Baffled ( θ Vertical-Baffled ( θ (g Force-Baffled ( θ =4 Force-Baffled ( θ F/(.5ρ gd θ F/(.5ρ gd θ F max1 =9.8 F max = F max1 =6.31 F max = M/(.5ρ gd 3 θ M max1 =5.9 M max =5.36 (d Moment ( θ =4 Moment ( θ M/(.5ρ gd 3 θ M max1 =3.3 M max =4.4 (h Moment-Baffled ( θ =4 Moment-Baffled ( θ Figure 15. The Comparison of the Unbaffled and Baffled Cases with the Various Parameters. ( ω = 1. rad/sec, β =1, d=ft 36