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Numerical simulation of 2D sloshing waves due to horizontal and vertical random excitation
Applied Ocean Research 28 (2006) 19–32 www.elsevier.com/locate/apor
Numerical simulation of 2D sloshing waves due to horizontal and vertical random excitation V. Sriram, S.A. Sannasiraj ∗ , V. Sundar Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600 036, India Received 17 October 2005; received in revised form 13 January 2006; accepted 14 January 2006 Available online 5 June 2006
Abstract The motion of sloshing waves under random excitation in the sway and heave modes have been simulated in a numerical wave tank. The fully nonlinear wave is numerically simulated using the finite element method with the cubic spline and finite difference approximations, in which the need for smoothing and regridding is minimal. The present model predictions are compared with that of Frandsen [Frandsen JB. Sloshing motions in the excited tanks. J Comput Phys 2004;196:53–87] for the regular wave excitation in the vertical and horizontal modes. The sloshing due to the simulated random excitation with different peak frequencies relative to the natural sloshing frequency has been subjected to frequency domain analysis. The results showed that irrespective of peak excitation frequency, the peaks appear at the natural frequencies of the system and the peak magnitude appears close to the natural frequency for the sway excitation. The higher magnitude is seen when the excitation frequency is equal to the first mode of natural frequency, due to the resonance condition. In the case of heave excitation, even though the peaks appear at the natural frequencies, the magnitude of the spectral peak remains the same for different excitation frequencies. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Sway; Heave; Fully nonlinear wave; Finite element method; Cubic spline approximation
1. Introduction Sloshing is a phenomenon which still needs considerable understanding in its behavior related to various engineering problems and is a composition of highly nonlinear waves that may lead to structural damage of the side walls of the container and/or may destabilize the ship. Numerous studies have been carried out in understanding sloshing waves using analytical, experimental and numerical approaches. The numerical approach has been carried out using twodimensional and three- dimensional tanks. In the case of two-dimensional tanks, Faltinsen [5,6] derived the analytical solution for sway and roll/pitch motions using the perturbation approach; Okamoto and Kawahara [19] and Armenio and La Rocca [1] compared the results of numerical and experimental data; Nakayama and Washizu [18] presented a numerical approach for forced pitching oscillation of the liquid tank;
∗ Corresponding author. Tel.: +91 44 2257 4817; fax: +91 44 2257 4802/4801. E-mail address: [email protected] (S.A. Sannasiraj).
c 2006 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2006.01.002
and Frandsen [10] analysed the sloshing motions in the vertical, horizontal and combined motions of the tank using analytical and numerical approaches. For the case of the threedimensional tank, Wu et al. [24] numerically simulated the sloshing waves and Huang and Hsiung [12] used the shallow water equations for the flow on the ship deck. Recently, Faltinsen et al. [8] performed the experimental investigation in a three-dimensional tank and compared the experimental results with the asymptotic modal system for longitudinal and diagonal excitations. Steady state regime for diagonal excitations versus water depth, excitation frequency and amplitude are classified. In this paper, the sloshing analysis is carried out for two types of excitatory motion, viz., horizontal and vertical excitations corresponding to the sway and heave modes of ship motion. The waves generated by the vertical excitation are called Faraday waves which were originally explored by Faraday [9] through experiments. Faraday waves are the resonant waves when the excitation frequency is twice the natural frequency for some initial perturbation in the container. This resonance condition is called parametric resonance. It is difficult to simulate sloshing in the experiments only with
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the vertical excitation. To have an initial perturbation in the free surface inside the container, horizontal motions need to be excited before the vertical excitation. A detailed review on sloshing of Faraday waves has been reported by Miles and Henderson [17]. Jiang et al. [15] and Bredmose et al. [3] have carried out experiments for the Faraday waves. A considerable amount of work has been carried out based on sloshing due to the horizontal excitation. The focus of many studies has been on the earthquake induced sloshing. Ibrahim et al. [13] gives a comprehensive review on sloshing motion predictions with more than 1000 references. Most of the work related to sloshing waves is based on the regular excitation of the container. This is quite useful for understanding the physical phenomenon of sloshing waves. However, these waves are not regular but random in nature. Recently, Wang and Khoo [22] have investigated the sloshing in a container due to random excitation in the horizontal directions. This work explored the knowledge towards the analysis for random excitation. The need for numerical modeling arises due to the significant importance of the higher order effects for the sloshing waves which are nonlinear, so neither the linear nor the second order potential considerations are sufficient enough to describe steep waves [10]. The advantage of numerical modelling is the flexibility in the simulation of the real sea state situation that has been carried out herein. The numerical approach of the present paper is similar to that of the formulation adopted by Wu and Eatock Taylor [23] in two dimensional numerical wave tank simulations. Herein, the finite element method (FEM) is used to find out the velocity potential, whereas, for the recovery of the velocity, a new formulation based on a cubic spline and finite difference approach has been adopted. The main advantage of this modification is that it reduces the need for smoothing or regridding for small amplitude waves. Sriram et al. [20] gives the details of its validity with the error estimation for the present scheme. The viscous effects play an important role after a critical depth [7]. Based on the dimensions of the tank, in the case of horizontal excitation, Wu et al. [24] reported the transition from standing wave form to progressive waves in the form of a bore. Hence, the dimensions of the tank are the influential parameters in the study of sloshing waves. As this paper mainly deals with the effect of random horizontal and vertical excitations, the dimension of the tank is assumed to be of constant length (L) of 2 m and depth (h) of 1 m, such that the depth aspect ratio (h/L) is 0.5. The present numerical model is initially validated with the numerical work of Frandsen [10] for regular motions and then applied for the computation of the sloshing due to random excitation, which is elaborated in detail. It is worthwhile to note that the results presented herein give rise to more questions, which motivated the authors for further investigations. 2. Mathematical formulation Let O0 x0 z 0 be the fixed coordinate system and O x z be the moving coordinate system fixed with the tank. These two coordinate systems coincide with each other when the tank is at
rest. The origins of this system are at the left end of the tank wall at the free surface and pointing upwards in z direction. These two coordinates along with the prescribed boundary conditions in each coordinate are represented in Fig. 1(a) and (b). The displacements of the tank are governed by the directions of axes as, X t = [xt (t), z t (t)].
(1)
On the assumption that the fluid is governed by potential flow theory, the velocity potential φ satisfies the Laplace equation, ∇ 2φ = 0
in the fluid domain, Ω .
(2)
The component of the water particle velocity normal to the walls of the tank is equal to the tank velocity ∂φ = U.n ∂n
in the side walls, Γ B
(3)
t where U = dX dt is the velocity of the tank and n is outward normal to the tank walls. The dynamic and kinematic free surface boundary condition in the fixed coordinate system can be written as,
∂φ 1 + ∇φ.∇φ + gη0 = 0 ∂t 2 ∂φ ∂η0 ∂φ ∂η0 + . − = 0. ∂t ∂ x0 ∂ x0 ∂z 0
(4) (5)
The free surface motion can be described in the moving coordinate system as, ∂φ dX t 1 − ∇φ. + .∇φ.∇φ + g(η + z t ) = 0 ∂t dt 2 ∂η ∂φ ∂φ dxt ∂η dz t + − − − . =0 ∂t ∂x dt ∂x ∂z dt
(6) (7)
which is obtained by substituting Eqs. (8) and (9) in Eqs. (4) and (5), ∇x0 z 0 = ∇x z ∂ dX t ∂ = − .∇ ∂t x z ∂t dt xz
(8) (9)
0 0
on z = η, where η = η0 − z t is the free surface elevation in the moving system O x z. Now, let the velocity potential be decomposed into, φ = ϕ + xu + zw
(10)
i.e., the velocity potential in the fixed coordinates system contains the velocity potential in the moving coordinates and the direction of the excitation with the corresponding velocity in that direction. where u and w are the velocity components in the x and z directions. Substituting Eq. (10) in Eqs. (2), (3), (6) and (7) leads to, ∇ 2 ϕ = 0 in the fluid domain, Ω ∂ϕ = 0 in the side walls, Γ B . ∂n
(11) (12)
V. Sriram et al. / Applied Ocean Research 28 (2006) 19–32
(a) Fixed coordinate system.
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(b) Moving coordinate system. Fig. 1. Sloshing wave tank modal domain.
The dynamic and kinematic free surface condition becomes, ∂ϕ 1 + .∇ϕ.∇ϕ + gη + x x T00 + ηz 00T = 0 ∂t 2 ∂η ∂ϕ ∂η ∂ϕ + . − = 0. ∂t ∂x ∂x ∂z
(13) (14)
Rewriting the above equations in Lagrangian form of motion following [16], dϕ 1 = ∇ϕ.∇ϕ − (g + z 00T (t))η − x x T00 (t) dt 2 dx ∂ϕ dz ∂ϕ = and = dt ∂x dt ∂z
(15) (16) Fig. 2. A typical mesh structure of the computational domain using three noded triangular element.
du 00 where z 00T (t) = dw dt , x T (t) = dt . The initial condition of the system can be assumed as,
φ(x0 , z 0 , t) = 0 η0 (x0 , t) = ζ0 where t = z 0 = 0;
(17) (18)
In moving coordinates, at t = 0 and z = 0, the above equations become, ϕ(x, 0, 0) = −xu η(x, 0) = 0 for horizontal excitation η(x, 0) = ζ0 for vertical excitation
governing Laplace’s equation constrained with the associated boundary conditions would lead to the following finite element systems of the equation [20], Z m X ∇ Ni ϕ j ∇ N j dΩ | j,i6∈Γs Ω
(19) (20a) (20b)
where ζ0 is the non-physical condition, specifying the initial elevation. Based on the initial condition, the boundary value problem is solved and the free surface elevation and potential values are updated at the subsequent time steps. The Eq. (12), (15) and (16), form the boundary conditions. Thus, this form of boundary conditions reduces the computational burden of creating a finer mesh structure along the input boundary, since the tank excitation is incorporated in the kinematic and dynamic free surface boundary condition. 3. Numerical procedure 3.1. FEM formulation The solution for the above boundary value problem is sought using a finite element scheme in this paper. Formulating the
j=1
Z ∇ Ni
=− Ω
m X
ϕ j ∇ N j dΩ | j∈Γs ,i6∈Γs
(21)
j=1
where ‘m’ is the total number of nodes in the domain and the potential [ϕ(x, z)] inside an element can be expressed in terms of its nodal potentials as ϕ j ϕ(x, z) =
n X
ϕ j N j (x, z).
(22)
j=1
Herein, N j is the shape function and n is the number of nodes in an element. The above formulation does not have any singularity effect at the intersection point between the free surface and the boundaries, compared to other methods like BEM or BIEM [23]. A linear triangular element is used. The mesh structure is shown in Fig. 2. 3.2. Cubic spline and finite difference approximations The updating of the free surface elevation poses a difficult problem in time domain simulation. Free surface smoothing or regridding are considered as alternatives by
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several investigators [11,24,22]. However, frequent smoothing or regridding would lead to non-conservation of energy and there is a need to limit the approximation procedures. In the present study, the horizontal velocity is calculated by fitting a cubic spline for ‘x’-coordinates and φ(x, z) values. The end conditions are considered as the natural spline condition. To evaluate the derivative at the ith node, five nodes are considered (two nodes on either side of the ith node), in order to minimize the effect of boundary constraints (natural spline condition). It ensures the smooth first derivatives. The computational time is also found to be efficient. Consider f i , f i0 and f i00 are continuous over the given interval. Based on the continuity condition, δxi 00 δxi + δxi+1 00 δxi+1 00 f i−1 + fi + f i+1 6 3 6 1 1 ( f i+1 − f i ) − = ( f i − f i−1 ) δxi+1 δxi i = 2, 3 . . . k − 1.
(23)
The above equation leads to a set of (k − 2) linear equations for the ‘k’ unknowns and δx is the horizontal spacing between the two nodes. It is solved by using the tridiagonal system of matrix assuming the second derivatives at the ends are zero i.e., the natural spline condition. In the present simulation, assuming f i = ϕi , the derivatives at a particular node (ϕi ) is found out by taking two nodes on either side (k = 5) with the second 00 , ϕ 00 ) at the end nodes being zero. After the derivatives (ϕi−2 i+2 evaluation of the second derivatives, the first derivatives can be estimated using the following equation, which are derived in the intermittent steps of the cubic spline interpolation [14] f i+1 − f i 6 00 − f i0 . (24) 2 f i00 + f i+1 = δxi δxi On the other hand, the vertical velocity can be estimated based on the backward finite difference (FD) scheme taking the advantages of distributing the nodes in the vertical line during mesh generation [21]. Consider ϕi ’s are the velocity potentials at the nodes corresponding to z i , where i = 1, 2, 3. Then, the vertical velocity at the free surface node can be obtained as, ∂ϕ (α 2 − 1)ϕ1 − α 2 ϕ2 + ϕ3 = ∂z α(α − 1)(z 1 − z 2 ) where α =
(25)
z 1 −z 3 z 1 −z 2 .
The algorithm for the numerical procedure is as follows, (1) Assume the initial velocity potential and surface elevation using Eqs. (19) and (20). (2) The entire domain is subdivided into finite elements and the nodes are numbered with proper element connectivity. (3) Using Eq. (21) solve for the velocity potential inside the domain. (4) Recover the velocity using the cubic spline and FD approximations in the free surface and check for the requirement of regridding. (5) Update the free surface nodes using Eqs. (15) and (16), the integration is carried out using the fourth order
Runge–Kutta method which require the evaluation of steps 3 and 4 three more times to obtain the new position. (6) After finding out the new free surface position and velocity potential perform step 2 till the termination time is reached. 4. Input generation 4.1. Regular wave excitation The container is assumed to take the following horizontal and vertical oscillations, x T (t) = ah H (ωh t) z T (t) = av V (ωv t)
(26a) (26b)
where ah,v and ωh,v are the characteristic amplitude and frequency of the wave. The motion of the container is assumed as V (χ ) = H (χ ) = cos(χ ). For the rectangular container, the order of the natural frequency is [5] p (27) ωn = gkn tanh(kn h) n = 1, 2, 3 . . . where the wave number is given by kn = nπ/L, L is the length of the container and h is the water depth. 4.2. Random wave excitation In this paper, the Bretschneider spectrum is used as the input wave spectrum which is given by, 5Hs2 ω p 5 5 ω p 4 . (28) Sη (ω) = exp − 16ω p ω 4 ω The free surface elevation is described by, η=
Nw X
Ai cos(ωi t + ψ i )
(29)
i=1
p where Ai is the amplitude that is defined by Ai = 2Sη (ω)1ω. Nw is the number of sinusoidal wave components, ωi and ψ i are the frequency and the phase angle, wherein the frequency ranges from 0 to π/dt and phase angle (a random variable based on the fixed seed number is used in this paper) ranges from 0 to 2π . Herein, ωi and Hs are the peak frequency and significant wave height. In this paper, the range of frequency has been taken up to the cutoff frequency that is assumed to be five times the natural frequency of the container; since the higher frequency will not have much influence on the generated waves. Based on the spectrum, the random waves are generated which are given as the input to the oscillation of the container; thus, Eq. (29) is assumed to be x T (t) =
Nw X
Ai cos(ωi t + ψ i )
i=1
for horizontal random oscillation Nw X z T (t) = Ai cos(ωi t + ψ i )
(30)
i=1
for vertical random oscillation.
(31)
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Fig. 3. Free surface elevation at the left wall due to regular horizontal excitation of frequencies, ωh = 0.7ω1 . (a) ah ωh2 = 0.0036g; (b) ah ωh2 = 0.036g. (— present numerical, Frandsen [10] numerical).
•••
Fig. 4. Power spectra of waves for the free surface elevation at the left corner of the wall due to horizontal excitation, ah = 0.005h (T1 = 2π/ω1 ).
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Fig. 5. Generated mesh due to horizontal excitation after every 5 s interval for ah = 0.005h and ωh = ω1 , number of nodes in x direction is 31 and in z direction is 13.
5. Results and discussion 5.1. Regular wave excitation 5.1.1. Horizontal excitation The numerical simulation of regular waves using the present methodology is compared with the numerical results of Frandsen [10]. The displacement of the container is given by the Eq. (26a) results in the excitation velocity as −ah ωh sin(ωh t). This leads the initial condition for velocity potential (Eq. (19)) to zero and the surface elevation is considered as in Eq. (20a) for this mode of excitation. For only horizontal excitation to exist, the vertical acceleration z 00T (t) is assumed to be zero. Comparisons have been made for a smaller and a steeper wave with the numerical simulation of Frandsen [10] (Fig. 3). It can be observed that both simulations are in close agreement for the forcing frequency (ωh ) equal to seventy percent of the first mode of natural frequency (ω1 ). There are 31 nodes in the x direction (free surface) and 13 nodes in the z direction and the time step adopted is 0.01 s for both simulations. An automatic regridding condition as proposed by Dommermuth and Yue [4] is adopted in the present model when the movement of the nodes is 75% more or less than the initial grid spacing. Smoothing is not adopted in the present simulation. It is found that no regridding is required for the simulation of small amplitude waves. In the case of steep waves, the need for regridding arises about 20 times for a simulated duration of 61.44 s. The induced sloshing time series is subjected to frequency domain analysis for a range of excitation frequencies with excitation amplitude of ah = 0.005h from which the occurrence of spectral peaks are identified. The spectral
density of free surface elevation in the tank for the excitation frequencies, ωh = 0.35ω1 , ωh = 0.75ω1 , ωh = ω1 , ωh = 1.5ω1 , ωh = 2.0ω1 and ωh = ω3 , are depicted in Fig. 4. The maximum spectral peak occurs at the excitation frequency when the excitation frequency is less than the natural frequency and a secondary peak occurs at the container natural frequency as can be seen in Fig. 4(a) and (b). While the excitation frequency is more than the container natural frequency and up to the second modal frequency [ω1 < ωh < ω2 (=1.5ω1 )], the primary peak occurs at the first mode followed by a secondary peak at the third natural frequency (Fig. 4(d)). For excitation frequencies equal to or greater than 1.8ω1 (=ω3 ), the third modal frequency dominates the sloshing motion. From Fig. 4(c) and (f), the sloshing motion is more violent at the natural frequency of the container when the excitation frequency is equal to the first mode rather than at the third mode, which is a well-known resonance phenomenon. The typical moving mesh generated after every 5 s interval is shown in Fig. 5 for the above said resonance condition. In order to have an in-depth knowledge on the occurrence and the magnitude of the spectral peaks with respect to the excitation frequency, the above simulation is carried out for different excitation frequencies. The magnitude of the first, third and excitation mode from the spectral analysis is shown in Fig. 6 with respect to the ratio of excitation by the first mode of the container frequency (excitation frequency ratio). The figure is separated into three zoomed zones for the excitation frequency ratio between 0.2 to 1.1, 1.1 to 1.55 and 1.55 to 2.6 (Fig. 6(a)). When the excitation frequency is less than the first mode (ωh < ω1 ), the primary spectral peak is observed at the excitation frequency as inferred from the results reported in
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Fig. 6. Bar chart showing the magnitude of different spectral peaks of sloshing waves due to horizontal excitation with frequencies (ω2 = 1.5ω1 , ω3 = 1.8ω1 , ω5 = 2.334ω1 ) (For interpretation of color legends in the figure, reader can refer the online version of this paper).
Fig. 6(b). The response component at the first modal frequency is observed to increase as the frequency increases. When the excitation frequency is greater than the first mode, the sloshing dominates at the first modal frequency up to second mode (ω2 = 1.5ω1 ) and till this frequency, the secondary peak is observed at the excitation frequency as can be seen in Fig. 6(c). When the excitation frequency is greater than ω1 , the domination of the first mode reduces with an increase in the frequency ratio. With a further increase in the excitation frequency ratio (ωh >= 1.5ω1 ), the spectral peak at third mode increases rapidly to 450 and 485 at ωh = 1.72ω1 and
ωh = 1.8ω1 respectively. This is clearly seen from Fig. 6(d). Furthermore, for the range of excitation frequency ratio 1.5 to 1.8, the response at the excitation frequency is insignificant. It is also observed that the sloshing is not dominant at the fifth or higher order modal frequency. Beyond the third modal frequency (ω3 ), the contribution at the excitation frequency is being influenced apart from oscillation at the first and third modal frequency components, but one should keep in mind the fifth mode (2.3ω1 ) of excitation also which is closer to these excitations. The maximum sloshing amplitude occurs when the
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Fig. 7. Free surface elevation at the left wall due to regular vertical excitation of frequency, ωv = 0.789ω1 , av ωv2 = 0.5g, (a) ε = 0.0014 (b) ε = 0.288, (— present numerical, Frandsen [10] numerical).
•••
Fig. 8. Power spectra of free surface sloshing waves at the left corner of the wall due to regular vertical excitation for an initial steepness of 0.014, av = 0.005h (T1 = 2π/ω1 ).
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Fig. 9. Power spectra of free surface sloshing waves at the left corner of the wall due to regular vertical excitation for an initial steepness of 0.288, av = 0.005h (T1 = 2π/ω1 ).
excitation is equal to the first mode and the next largest being 5.65 times lesser occurring at ω3 . 5.1.2. Vertical excitation To simulate the condition of vertical excitation for the container, x T00 is set to zero in Eqs. (13) and (15). The tank is assumed to be periodically excited with the displacement given by Eq. (26b), which leads to a velocity of the form −av ωv sin(ωv t). Thus, the initial velocity potential becomes zero according to Eq. (19). The initial condition for the surface elevation is an important parameter as there should be some initial perturbation in the system for the generation of waves due to vertical excitation. The linear solution and the stability criteria of Faraday waves are given by Benjamin and Ursell [2]. In the numerical simulation, it is quite usual to adopt an initial free surface perturbation of η = ζ0 = a cos(kn x). It should be noted that this condition does not necessarily arise in
the real situation because both horizontal and vertical motions co-exist. For experimental purposes, the initial perturbation is introduced in the system, by exciting the tank horizontally for the prescribed time. Two typical sloshing simulations are carried out, following small amplitude motion and steep wave excitation as shown in Fig. 7. The steepness parameter depends on the adopted initial condition, ε = aωn2 /g. The present simulation shows a close agreement with the numerical simulation of Frandsen [10]. The grid size and the time step adopted are the same as that for horizontal motion. A frequency domain analysis for the sloshing motion due to vertical excitation has been carried out. Even though it may not clearly depict the exact phenomenon of the waves such as the detonating effects as noticed by Frandsen [10], it gives the picture of the critical resonance condition based on the intensity as discussed in sway mode. The spectra of sloshing time series for two different initial steepness conditions of 0.014
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Fig. 10. (a) Typical excitation spectrum with Hs = 0.01h and ω p = ω1 . (b) Displacement generated from the spectrum. (c) Free surface elevation at the left corner of the wall due to horizontal motions prescribed by (b). (d) Free surface elevation at the left corner of the wall due to vertical motions prescribed by (b).
and 0.288 with constant excitation amplitude are presented in Figs. 8 and 9 respectively. Such plots have been reported for ωv = 0.75ω1 , ω1 , 1.5ω1 , 2ω1 and 2.5ω1 . A comparison of the results presented in the above two figures reveals the following. In the case of lesser initial steepness, a single peak of the dimensionless sloshing energy is observed at frequency ratio of 1, irrespective of the magnitude of the excitation frequency. In the case of the higher initial steepness adopted in the study, the dimensionless sloshing frequency spectra exhibits secondary peaks at ω/ω1 = 1.5 and 2 for all the ωv tested. The magnitude of the secondary peak is insignificant compared to the primary peak at the resonance frequency. Frandsen [10] noticed a similar existence of peaks in the wave spectra. 5.2. Random wave excitation 5.2.1. General In order to understand the sloshing phenomenon, the container is subjected to random excitation under the real random sea state, with different peak frequency by keeping the total supplied energy to the system as constant. A typical input excitation spectrum and the corresponding displacement time history are given in Fig. 10(a) and (b) respectively. The horizontal and vertical container displacements are obtained from this spectrum (Eqs. (30) or (31)). The surface elevations at the left wall of the container due to horizontal (η/ h) H and vertical (η/ h)V excitation are shown in Fig. 10(c) and (d) respectively. The time step adopted is 0.06 s and the duration of the simulation is 61.44 s corresponding to 1024 data points, which is used to analyse the spectrum herein.
5.2.2. Horizontal excitation For a significant wave height of 0.006h, the tank is excited with random waves of different peak frequency (ω p = 0.35ω1 , ω p = 0.75ω1 , ω p = ω1 , ω p = 1.5 ω1 , ω p = ω3 ). The power spectra of free surface sloshing elevation at the left corner of the tank wall for various excitation peak frequencies are shown in Fig. 11. It can be seen that while the excitation peak frequency is less than the natural frequency, the sloshing spectral peaks appear only at natural frequencies (first, third and fifth mode) of the tank and no peak is visible at the excitation peak frequency as noticed in the case of regular excitation. The primary spectral peak lies at the first mode. When the excitation peak frequency is greater than the first mode of the tank system, there were some high frequency waves in the container (Fig. 11(d), (e) and (f)). Thus, in horizontal excitation, the spectral peaks appear only at the first or higher mode natural frequency, irrespective of the excitation peak frequency. A similar trend in the occurrence of spectral peaks is also noticed by Wang and Khoo [22]. The spectrum for forces and surface elevation of the tank wall was examined using FEM adopting isoparametric elements. It has been shown that the energy mainly concentrates at the natural frequency of the container and is found to dominate at the ith mode of the container when the peak frequency is close to the ith mode. 5.2.3. Vertical excitation The sloshing behavior of the tank when subjected to random vertical excitation is studied. The container was subjected to
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Fig. 11. Spectra of free surface sloshing waves at the left corner of the wall due to horizontal random excitation (Hs = 0.006h, T1 = 2π/ω1 ).
vertical excitation with the same input characteristics as that for horizontal motion for two different initial conditions. The initial conditions correspond to small amplitude with a steepness of 0.014 and large amplitude with a steepness of 0.288. The power spectra of waves at the left corner of the tank wall with the above initial conditions having steepness of 0.014 and 0.288 are shown in Figs. 12 and 13 respectively. It can be seen that inspite of the excitation frequency, the spectral peaks appear at the first modal frequency only while the initial perturbed waves are of small steepness. The spectral peaks appear at first, second and two times the first mode (parametric) frequency, if the initial perturbed waves are of large steepness, which is similar to that of the regular motion as discussed earlier for the heave mode. The dominating frequency appears at the first mode. However, the spectral peak magnitudes at the first modal frequency are of the same order, irrespective of excitation peak frequency. This phenomenon is observed irrespective of the different initial
perturbation in the tank. It should be noted that no regridding is necessary for the small steepness case, and hence the numerical damping does not play any role for the above results. However, in the case of regular motion, the magnitude of peak is higher at ωv = 2ω1 for small and high initial steepness. 6. Summary and conclusion This paper presents the behavior of the sloshing waves in a tank subjected to excitation in the horizontal and vertical directions using the inviscid flow solver. Both regular and random excitations have been considered. The salient conclusions drawn from the study are presented below. Due to regular excitation in the horizontal direction, the dominating spectral peak appears at the excitation mode, when the excitation frequency is less than the first mode, beyond which the sloshing is dominant at the natural frequency of the
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Fig. 12. Spectra of free surface sloshing waves at the left corner of the wall due to vertical random excitation for an initial steepness of 0.014 and Hs = 0.006h (T1 = 2π/ω1 ).
container. The critical sloshing motion occurs at the well known resonance condition, while the excitation frequency is equal to the first mode. Due to regular excitation in the vertical direction, the spectral peak appears to be at the first mode for small initial perturbation and at first, second and two times of the first mode for higher initial perturbation for constant excitation amplitude. The dominating spectral peaks in both the cases are at the first mode. The critical sloshing occurs at the excitation frequency equal to twice the first mode, well known as the parametric resonance. For the random excitation in the horizontal direction, the spectral peaks occur only at the natural frequencies of the container, the dominating peak appear close to ith mode, when the excitation peak frequency is at ith mode. The maximum intensity appears to be at the excitation peak frequency equal to the first mode that is, resonance condition.
In the case of random excitation in vertical direction, the dominating peak appears only at the first mode. The magnitude of the peak is almost the same irrespective of the excitation peak frequency and initial perturbation, contrary to the regular excitation, wherein the magnitude is large only if the excitation frequency is equal to twice the first mode (parametric resonance) irrespective of initial perturbation. Frandsen [10] showed that the combined effect of vertical and horizontal regular excitations lead to an infinite number of additional resonant frequency components. The numerical results of the individual excitation in horizontal and vertical directions are in good agreement with the results of Frandsen [10]. The magnitude of the peak is almost the same irrespective of the excitation peak frequency and initial perturbation for vertical random excitation. This requires further investigation in random excitation to examine the effect of the additional frequency resonance for combined random excitation.
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Fig. 13. Spectra of free surface sloshing waves at the left corner of the wall due to vertical random excitation for an initial steepness of 0.288 and Hs = 0.006h (T1 = 2π/ω1 ).
References [1] Armenio V, La Rocca M. On the analysis of sloshing of water in rectangular containers: Numerical study and experimental validation. Ocean Eng 1996;23(8):705–39. [2] Benjamin TB, Ursell F. The stability of the plane free surface of a liquid in a vertical periodic motion. Proc R Soc Lond Ser A 1954;225:505–15. [3] Bredmose H, Brocchini M, Peregrine DH, Thais L. Experimental investigation and numerical modelling of steep forced water waves. J Fluid Mech 2003;490:217–49. [4] Dommermuth DG, Yue DKP. Numerical simulation of nonlinear axisymmetric flows with a free surface. J Fluid Mech 1987;178:195–219. [5] Faltinsen OM. A nonlinear theory of sloshing in rectangular tanks. J Ship Res 1974;18(4):224–41. [6] Faltinsen OM. A numerical non-linear method of sloshing in tanks with two dimensional flow. J Ship Res 1978;18(4):224–41. [7] Faltinsen OM, Rognebakke OF, Lukovsky IA, Timokha AN. Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J Fluid Mech 2000;407:201–34.
[8] Faltinsen OM, Rognebakke OF, Timokha AN. Classification of threedimensional nonlinear sloshing in a square-base tank with finite depth. J Fluids Struct 2005;20:81–103. [9] Faraday M. On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil Trans R Soc Lond 1831;121:299–340. [10] Frandsen JB. Sloshing motions in the excited tanks. J Comput Phys 2004; 196:53–87. [11] Greaves DM, Borthwick AGL, Wu GX, Eatock Taylor R. A moving boundary finite element method for fully non-linear wave simulations. J Ship Res 1997;41(3):181–94. [12] Huang ZJ, Hsiung CC. Nonlinear shallow-water flow on deck. J Ship Res 1996;40(4):303–15. [13] Ibrahim RA, Pilipchuk VN, Ikeda T. Recent advances in liquid sloshing dynamics. ASME Appl Mech Rev 2001;54(2):133–99. [14] Jain MK, Iyengar SRK, Jain RK. Numerical methods for scientific and engineering computation. New Delhi: New Age International Publishers; 2003.
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[15] Jiang L, Ting C-L, Perlin M, Schultz WW. Moderate and steep faraday waves: Instabilities, modulation and temporal asymmetries. J Fluid Mech 1996;329:275–307. [16] Longuet-Higgins MS, Cokelet ED. The deformation of steep surface waves on water: I. A numerical method of computation. Proc R Soc Lond A 1976;350:1–26. [17] Miles J, Henderson D. Parametrically forced surface waves. Ann Rev Fluid Mech 1990;22:143–65. [18] Nakayama T, Washizu K. Nonlinear analysis of liquid motion in a container subjected to forced pitching oscillation. Internat J Numer Methods Engrg 1980;15:1207–20. [19] Okamoto T, Kawahara M. Two-dimensional sloshing analysis by Lagrangian finite element method. Internat J Numer Methods Fluids 1990; 11:453–77.
[20] Sriram V, Sannasiraj SA, Sundar V. Numerical simulation of 2D nonlinear waves using finite element with cubic spline approximations. J Fluids Struct 2006;22(5):663–81. [21] Turnbull MS, Borthwick AGL, Eatock Taylor R. Numerical wave tank based on sigma-transformed finite element inviscid flow solver. Internat J Numer Methods Fluids 2003;42:641–63. [22] Wang CZ, Khoo BC. Finite Element analysis of two-dimensional nonlinear sloshing problems in random excitation. Ocean Eng 2005;32: 107–33. [23] Wu GX, Eatock Taylor R. Finite element analysis of two-dimensional non-linear transient water waves. Appl Ocean Res 1994;16:363–72. [24] Wu GX, Ma QW, Eatock Taylor R. Numerical simulation of sloshing waves in a 3D tank based on a finite element method. Appl Ocean Res 1998;20:337–55.
Numerical simulation of 2D sloshing waves due to horizontal and vertical random excitation V. Sriram, S.A. Sannasiraj ∗ , V. Sundar Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai 600 036, India Received 17 October 2005; received in revised form 13 January 2006; accepted 14 January 2006 Available online 5 June 2006
Abstract The motion of sloshing waves under random excitation in the sway and heave modes have been simulated in a numerical wave tank. The fully nonlinear wave is numerically simulated using the finite element method with the cubic spline and finite difference approximations, in which the need for smoothing and regridding is minimal. The present model predictions are compared with that of Frandsen [Frandsen JB. Sloshing motions in the excited tanks. J Comput Phys 2004;196:53–87] for the regular wave excitation in the vertical and horizontal modes. The sloshing due to the simulated random excitation with different peak frequencies relative to the natural sloshing frequency has been subjected to frequency domain analysis. The results showed that irrespective of peak excitation frequency, the peaks appear at the natural frequencies of the system and the peak magnitude appears close to the natural frequency for the sway excitation. The higher magnitude is seen when the excitation frequency is equal to the first mode of natural frequency, due to the resonance condition. In the case of heave excitation, even though the peaks appear at the natural frequencies, the magnitude of the spectral peak remains the same for different excitation frequencies. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Sway; Heave; Fully nonlinear wave; Finite element method; Cubic spline approximation
1. Introduction Sloshing is a phenomenon which still needs considerable understanding in its behavior related to various engineering problems and is a composition of highly nonlinear waves that may lead to structural damage of the side walls of the container and/or may destabilize the ship. Numerous studies have been carried out in understanding sloshing waves using analytical, experimental and numerical approaches. The numerical approach has been carried out using twodimensional and three- dimensional tanks. In the case of two-dimensional tanks, Faltinsen [5,6] derived the analytical solution for sway and roll/pitch motions using the perturbation approach; Okamoto and Kawahara [19] and Armenio and La Rocca [1] compared the results of numerical and experimental data; Nakayama and Washizu [18] presented a numerical approach for forced pitching oscillation of the liquid tank;
∗ Corresponding author. Tel.: +91 44 2257 4817; fax: +91 44 2257 4802/4801. E-mail address: [email protected] (S.A. Sannasiraj).
c 2006 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2006.01.002
and Frandsen [10] analysed the sloshing motions in the vertical, horizontal and combined motions of the tank using analytical and numerical approaches. For the case of the threedimensional tank, Wu et al. [24] numerically simulated the sloshing waves and Huang and Hsiung [12] used the shallow water equations for the flow on the ship deck. Recently, Faltinsen et al. [8] performed the experimental investigation in a three-dimensional tank and compared the experimental results with the asymptotic modal system for longitudinal and diagonal excitations. Steady state regime for diagonal excitations versus water depth, excitation frequency and amplitude are classified. In this paper, the sloshing analysis is carried out for two types of excitatory motion, viz., horizontal and vertical excitations corresponding to the sway and heave modes of ship motion. The waves generated by the vertical excitation are called Faraday waves which were originally explored by Faraday [9] through experiments. Faraday waves are the resonant waves when the excitation frequency is twice the natural frequency for some initial perturbation in the container. This resonance condition is called parametric resonance. It is difficult to simulate sloshing in the experiments only with
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the vertical excitation. To have an initial perturbation in the free surface inside the container, horizontal motions need to be excited before the vertical excitation. A detailed review on sloshing of Faraday waves has been reported by Miles and Henderson [17]. Jiang et al. [15] and Bredmose et al. [3] have carried out experiments for the Faraday waves. A considerable amount of work has been carried out based on sloshing due to the horizontal excitation. The focus of many studies has been on the earthquake induced sloshing. Ibrahim et al. [13] gives a comprehensive review on sloshing motion predictions with more than 1000 references. Most of the work related to sloshing waves is based on the regular excitation of the container. This is quite useful for understanding the physical phenomenon of sloshing waves. However, these waves are not regular but random in nature. Recently, Wang and Khoo [22] have investigated the sloshing in a container due to random excitation in the horizontal directions. This work explored the knowledge towards the analysis for random excitation. The need for numerical modeling arises due to the significant importance of the higher order effects for the sloshing waves which are nonlinear, so neither the linear nor the second order potential considerations are sufficient enough to describe steep waves [10]. The advantage of numerical modelling is the flexibility in the simulation of the real sea state situation that has been carried out herein. The numerical approach of the present paper is similar to that of the formulation adopted by Wu and Eatock Taylor [23] in two dimensional numerical wave tank simulations. Herein, the finite element method (FEM) is used to find out the velocity potential, whereas, for the recovery of the velocity, a new formulation based on a cubic spline and finite difference approach has been adopted. The main advantage of this modification is that it reduces the need for smoothing or regridding for small amplitude waves. Sriram et al. [20] gives the details of its validity with the error estimation for the present scheme. The viscous effects play an important role after a critical depth [7]. Based on the dimensions of the tank, in the case of horizontal excitation, Wu et al. [24] reported the transition from standing wave form to progressive waves in the form of a bore. Hence, the dimensions of the tank are the influential parameters in the study of sloshing waves. As this paper mainly deals with the effect of random horizontal and vertical excitations, the dimension of the tank is assumed to be of constant length (L) of 2 m and depth (h) of 1 m, such that the depth aspect ratio (h/L) is 0.5. The present numerical model is initially validated with the numerical work of Frandsen [10] for regular motions and then applied for the computation of the sloshing due to random excitation, which is elaborated in detail. It is worthwhile to note that the results presented herein give rise to more questions, which motivated the authors for further investigations. 2. Mathematical formulation Let O0 x0 z 0 be the fixed coordinate system and O x z be the moving coordinate system fixed with the tank. These two coordinate systems coincide with each other when the tank is at
rest. The origins of this system are at the left end of the tank wall at the free surface and pointing upwards in z direction. These two coordinates along with the prescribed boundary conditions in each coordinate are represented in Fig. 1(a) and (b). The displacements of the tank are governed by the directions of axes as, X t = [xt (t), z t (t)].
(1)
On the assumption that the fluid is governed by potential flow theory, the velocity potential φ satisfies the Laplace equation, ∇ 2φ = 0
in the fluid domain, Ω .
(2)
The component of the water particle velocity normal to the walls of the tank is equal to the tank velocity ∂φ = U.n ∂n
in the side walls, Γ B
(3)
t where U = dX dt is the velocity of the tank and n is outward normal to the tank walls. The dynamic and kinematic free surface boundary condition in the fixed coordinate system can be written as,
∂φ 1 + ∇φ.∇φ + gη0 = 0 ∂t 2 ∂φ ∂η0 ∂φ ∂η0 + . − = 0. ∂t ∂ x0 ∂ x0 ∂z 0
(4) (5)
The free surface motion can be described in the moving coordinate system as, ∂φ dX t 1 − ∇φ. + .∇φ.∇φ + g(η + z t ) = 0 ∂t dt 2 ∂η ∂φ ∂φ dxt ∂η dz t + − − − . =0 ∂t ∂x dt ∂x ∂z dt
(6) (7)
which is obtained by substituting Eqs. (8) and (9) in Eqs. (4) and (5), ∇x0 z 0 = ∇x z ∂ dX t ∂ = − .∇ ∂t x z ∂t dt xz
(8) (9)
0 0
on z = η, where η = η0 − z t is the free surface elevation in the moving system O x z. Now, let the velocity potential be decomposed into, φ = ϕ + xu + zw
(10)
i.e., the velocity potential in the fixed coordinates system contains the velocity potential in the moving coordinates and the direction of the excitation with the corresponding velocity in that direction. where u and w are the velocity components in the x and z directions. Substituting Eq. (10) in Eqs. (2), (3), (6) and (7) leads to, ∇ 2 ϕ = 0 in the fluid domain, Ω ∂ϕ = 0 in the side walls, Γ B . ∂n
(11) (12)
V. Sriram et al. / Applied Ocean Research 28 (2006) 19–32
(a) Fixed coordinate system.
21
(b) Moving coordinate system. Fig. 1. Sloshing wave tank modal domain.
The dynamic and kinematic free surface condition becomes, ∂ϕ 1 + .∇ϕ.∇ϕ + gη + x x T00 + ηz 00T = 0 ∂t 2 ∂η ∂ϕ ∂η ∂ϕ + . − = 0. ∂t ∂x ∂x ∂z
(13) (14)
Rewriting the above equations in Lagrangian form of motion following [16], dϕ 1 = ∇ϕ.∇ϕ − (g + z 00T (t))η − x x T00 (t) dt 2 dx ∂ϕ dz ∂ϕ = and = dt ∂x dt ∂z
(15) (16) Fig. 2. A typical mesh structure of the computational domain using three noded triangular element.
du 00 where z 00T (t) = dw dt , x T (t) = dt . The initial condition of the system can be assumed as,
φ(x0 , z 0 , t) = 0 η0 (x0 , t) = ζ0 where t = z 0 = 0;
(17) (18)
In moving coordinates, at t = 0 and z = 0, the above equations become, ϕ(x, 0, 0) = −xu η(x, 0) = 0 for horizontal excitation η(x, 0) = ζ0 for vertical excitation
governing Laplace’s equation constrained with the associated boundary conditions would lead to the following finite element systems of the equation [20], Z m X ∇ Ni ϕ j ∇ N j dΩ | j,i6∈Γs Ω
(19) (20a) (20b)
where ζ0 is the non-physical condition, specifying the initial elevation. Based on the initial condition, the boundary value problem is solved and the free surface elevation and potential values are updated at the subsequent time steps. The Eq. (12), (15) and (16), form the boundary conditions. Thus, this form of boundary conditions reduces the computational burden of creating a finer mesh structure along the input boundary, since the tank excitation is incorporated in the kinematic and dynamic free surface boundary condition. 3. Numerical procedure 3.1. FEM formulation The solution for the above boundary value problem is sought using a finite element scheme in this paper. Formulating the
j=1
Z ∇ Ni
=− Ω
m X
ϕ j ∇ N j dΩ | j∈Γs ,i6∈Γs
(21)
j=1
where ‘m’ is the total number of nodes in the domain and the potential [ϕ(x, z)] inside an element can be expressed in terms of its nodal potentials as ϕ j ϕ(x, z) =
n X
ϕ j N j (x, z).
(22)
j=1
Herein, N j is the shape function and n is the number of nodes in an element. The above formulation does not have any singularity effect at the intersection point between the free surface and the boundaries, compared to other methods like BEM or BIEM [23]. A linear triangular element is used. The mesh structure is shown in Fig. 2. 3.2. Cubic spline and finite difference approximations The updating of the free surface elevation poses a difficult problem in time domain simulation. Free surface smoothing or regridding are considered as alternatives by
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several investigators [11,24,22]. However, frequent smoothing or regridding would lead to non-conservation of energy and there is a need to limit the approximation procedures. In the present study, the horizontal velocity is calculated by fitting a cubic spline for ‘x’-coordinates and φ(x, z) values. The end conditions are considered as the natural spline condition. To evaluate the derivative at the ith node, five nodes are considered (two nodes on either side of the ith node), in order to minimize the effect of boundary constraints (natural spline condition). It ensures the smooth first derivatives. The computational time is also found to be efficient. Consider f i , f i0 and f i00 are continuous over the given interval. Based on the continuity condition, δxi 00 δxi + δxi+1 00 δxi+1 00 f i−1 + fi + f i+1 6 3 6 1 1 ( f i+1 − f i ) − = ( f i − f i−1 ) δxi+1 δxi i = 2, 3 . . . k − 1.
(23)
The above equation leads to a set of (k − 2) linear equations for the ‘k’ unknowns and δx is the horizontal spacing between the two nodes. It is solved by using the tridiagonal system of matrix assuming the second derivatives at the ends are zero i.e., the natural spline condition. In the present simulation, assuming f i = ϕi , the derivatives at a particular node (ϕi ) is found out by taking two nodes on either side (k = 5) with the second 00 , ϕ 00 ) at the end nodes being zero. After the derivatives (ϕi−2 i+2 evaluation of the second derivatives, the first derivatives can be estimated using the following equation, which are derived in the intermittent steps of the cubic spline interpolation [14] f i+1 − f i 6 00 − f i0 . (24) 2 f i00 + f i+1 = δxi δxi On the other hand, the vertical velocity can be estimated based on the backward finite difference (FD) scheme taking the advantages of distributing the nodes in the vertical line during mesh generation [21]. Consider ϕi ’s are the velocity potentials at the nodes corresponding to z i , where i = 1, 2, 3. Then, the vertical velocity at the free surface node can be obtained as, ∂ϕ (α 2 − 1)ϕ1 − α 2 ϕ2 + ϕ3 = ∂z α(α − 1)(z 1 − z 2 ) where α =
(25)
z 1 −z 3 z 1 −z 2 .
The algorithm for the numerical procedure is as follows, (1) Assume the initial velocity potential and surface elevation using Eqs. (19) and (20). (2) The entire domain is subdivided into finite elements and the nodes are numbered with proper element connectivity. (3) Using Eq. (21) solve for the velocity potential inside the domain. (4) Recover the velocity using the cubic spline and FD approximations in the free surface and check for the requirement of regridding. (5) Update the free surface nodes using Eqs. (15) and (16), the integration is carried out using the fourth order
Runge–Kutta method which require the evaluation of steps 3 and 4 three more times to obtain the new position. (6) After finding out the new free surface position and velocity potential perform step 2 till the termination time is reached. 4. Input generation 4.1. Regular wave excitation The container is assumed to take the following horizontal and vertical oscillations, x T (t) = ah H (ωh t) z T (t) = av V (ωv t)
(26a) (26b)
where ah,v and ωh,v are the characteristic amplitude and frequency of the wave. The motion of the container is assumed as V (χ ) = H (χ ) = cos(χ ). For the rectangular container, the order of the natural frequency is [5] p (27) ωn = gkn tanh(kn h) n = 1, 2, 3 . . . where the wave number is given by kn = nπ/L, L is the length of the container and h is the water depth. 4.2. Random wave excitation In this paper, the Bretschneider spectrum is used as the input wave spectrum which is given by, 5Hs2 ω p 5 5 ω p 4 . (28) Sη (ω) = exp − 16ω p ω 4 ω The free surface elevation is described by, η=
Nw X
Ai cos(ωi t + ψ i )
(29)
i=1
p where Ai is the amplitude that is defined by Ai = 2Sη (ω)1ω. Nw is the number of sinusoidal wave components, ωi and ψ i are the frequency and the phase angle, wherein the frequency ranges from 0 to π/dt and phase angle (a random variable based on the fixed seed number is used in this paper) ranges from 0 to 2π . Herein, ωi and Hs are the peak frequency and significant wave height. In this paper, the range of frequency has been taken up to the cutoff frequency that is assumed to be five times the natural frequency of the container; since the higher frequency will not have much influence on the generated waves. Based on the spectrum, the random waves are generated which are given as the input to the oscillation of the container; thus, Eq. (29) is assumed to be x T (t) =
Nw X
Ai cos(ωi t + ψ i )
i=1
for horizontal random oscillation Nw X z T (t) = Ai cos(ωi t + ψ i )
(30)
i=1
for vertical random oscillation.
(31)
V. Sriram et al. / Applied Ocean Research 28 (2006) 19–32
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Fig. 3. Free surface elevation at the left wall due to regular horizontal excitation of frequencies, ωh = 0.7ω1 . (a) ah ωh2 = 0.0036g; (b) ah ωh2 = 0.036g. (— present numerical, Frandsen [10] numerical).
•••
Fig. 4. Power spectra of waves for the free surface elevation at the left corner of the wall due to horizontal excitation, ah = 0.005h (T1 = 2π/ω1 ).
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Fig. 5. Generated mesh due to horizontal excitation after every 5 s interval for ah = 0.005h and ωh = ω1 , number of nodes in x direction is 31 and in z direction is 13.
5. Results and discussion 5.1. Regular wave excitation 5.1.1. Horizontal excitation The numerical simulation of regular waves using the present methodology is compared with the numerical results of Frandsen [10]. The displacement of the container is given by the Eq. (26a) results in the excitation velocity as −ah ωh sin(ωh t). This leads the initial condition for velocity potential (Eq. (19)) to zero and the surface elevation is considered as in Eq. (20a) for this mode of excitation. For only horizontal excitation to exist, the vertical acceleration z 00T (t) is assumed to be zero. Comparisons have been made for a smaller and a steeper wave with the numerical simulation of Frandsen [10] (Fig. 3). It can be observed that both simulations are in close agreement for the forcing frequency (ωh ) equal to seventy percent of the first mode of natural frequency (ω1 ). There are 31 nodes in the x direction (free surface) and 13 nodes in the z direction and the time step adopted is 0.01 s for both simulations. An automatic regridding condition as proposed by Dommermuth and Yue [4] is adopted in the present model when the movement of the nodes is 75% more or less than the initial grid spacing. Smoothing is not adopted in the present simulation. It is found that no regridding is required for the simulation of small amplitude waves. In the case of steep waves, the need for regridding arises about 20 times for a simulated duration of 61.44 s. The induced sloshing time series is subjected to frequency domain analysis for a range of excitation frequencies with excitation amplitude of ah = 0.005h from which the occurrence of spectral peaks are identified. The spectral
density of free surface elevation in the tank for the excitation frequencies, ωh = 0.35ω1 , ωh = 0.75ω1 , ωh = ω1 , ωh = 1.5ω1 , ωh = 2.0ω1 and ωh = ω3 , are depicted in Fig. 4. The maximum spectral peak occurs at the excitation frequency when the excitation frequency is less than the natural frequency and a secondary peak occurs at the container natural frequency as can be seen in Fig. 4(a) and (b). While the excitation frequency is more than the container natural frequency and up to the second modal frequency [ω1 < ωh < ω2 (=1.5ω1 )], the primary peak occurs at the first mode followed by a secondary peak at the third natural frequency (Fig. 4(d)). For excitation frequencies equal to or greater than 1.8ω1 (=ω3 ), the third modal frequency dominates the sloshing motion. From Fig. 4(c) and (f), the sloshing motion is more violent at the natural frequency of the container when the excitation frequency is equal to the first mode rather than at the third mode, which is a well-known resonance phenomenon. The typical moving mesh generated after every 5 s interval is shown in Fig. 5 for the above said resonance condition. In order to have an in-depth knowledge on the occurrence and the magnitude of the spectral peaks with respect to the excitation frequency, the above simulation is carried out for different excitation frequencies. The magnitude of the first, third and excitation mode from the spectral analysis is shown in Fig. 6 with respect to the ratio of excitation by the first mode of the container frequency (excitation frequency ratio). The figure is separated into three zoomed zones for the excitation frequency ratio between 0.2 to 1.1, 1.1 to 1.55 and 1.55 to 2.6 (Fig. 6(a)). When the excitation frequency is less than the first mode (ωh < ω1 ), the primary spectral peak is observed at the excitation frequency as inferred from the results reported in
V. Sriram et al. / Applied Ocean Research 28 (2006) 19–32
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Fig. 6. Bar chart showing the magnitude of different spectral peaks of sloshing waves due to horizontal excitation with frequencies (ω2 = 1.5ω1 , ω3 = 1.8ω1 , ω5 = 2.334ω1 ) (For interpretation of color legends in the figure, reader can refer the online version of this paper).
Fig. 6(b). The response component at the first modal frequency is observed to increase as the frequency increases. When the excitation frequency is greater than the first mode, the sloshing dominates at the first modal frequency up to second mode (ω2 = 1.5ω1 ) and till this frequency, the secondary peak is observed at the excitation frequency as can be seen in Fig. 6(c). When the excitation frequency is greater than ω1 , the domination of the first mode reduces with an increase in the frequency ratio. With a further increase in the excitation frequency ratio (ωh >= 1.5ω1 ), the spectral peak at third mode increases rapidly to 450 and 485 at ωh = 1.72ω1 and
ωh = 1.8ω1 respectively. This is clearly seen from Fig. 6(d). Furthermore, for the range of excitation frequency ratio 1.5 to 1.8, the response at the excitation frequency is insignificant. It is also observed that the sloshing is not dominant at the fifth or higher order modal frequency. Beyond the third modal frequency (ω3 ), the contribution at the excitation frequency is being influenced apart from oscillation at the first and third modal frequency components, but one should keep in mind the fifth mode (2.3ω1 ) of excitation also which is closer to these excitations. The maximum sloshing amplitude occurs when the
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Fig. 7. Free surface elevation at the left wall due to regular vertical excitation of frequency, ωv = 0.789ω1 , av ωv2 = 0.5g, (a) ε = 0.0014 (b) ε = 0.288, (— present numerical, Frandsen [10] numerical).
•••
Fig. 8. Power spectra of free surface sloshing waves at the left corner of the wall due to regular vertical excitation for an initial steepness of 0.014, av = 0.005h (T1 = 2π/ω1 ).
V. Sriram et al. / Applied Ocean Research 28 (2006) 19–32
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Fig. 9. Power spectra of free surface sloshing waves at the left corner of the wall due to regular vertical excitation for an initial steepness of 0.288, av = 0.005h (T1 = 2π/ω1 ).
excitation is equal to the first mode and the next largest being 5.65 times lesser occurring at ω3 . 5.1.2. Vertical excitation To simulate the condition of vertical excitation for the container, x T00 is set to zero in Eqs. (13) and (15). The tank is assumed to be periodically excited with the displacement given by Eq. (26b), which leads to a velocity of the form −av ωv sin(ωv t). Thus, the initial velocity potential becomes zero according to Eq. (19). The initial condition for the surface elevation is an important parameter as there should be some initial perturbation in the system for the generation of waves due to vertical excitation. The linear solution and the stability criteria of Faraday waves are given by Benjamin and Ursell [2]. In the numerical simulation, it is quite usual to adopt an initial free surface perturbation of η = ζ0 = a cos(kn x). It should be noted that this condition does not necessarily arise in
the real situation because both horizontal and vertical motions co-exist. For experimental purposes, the initial perturbation is introduced in the system, by exciting the tank horizontally for the prescribed time. Two typical sloshing simulations are carried out, following small amplitude motion and steep wave excitation as shown in Fig. 7. The steepness parameter depends on the adopted initial condition, ε = aωn2 /g. The present simulation shows a close agreement with the numerical simulation of Frandsen [10]. The grid size and the time step adopted are the same as that for horizontal motion. A frequency domain analysis for the sloshing motion due to vertical excitation has been carried out. Even though it may not clearly depict the exact phenomenon of the waves such as the detonating effects as noticed by Frandsen [10], it gives the picture of the critical resonance condition based on the intensity as discussed in sway mode. The spectra of sloshing time series for two different initial steepness conditions of 0.014
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Fig. 10. (a) Typical excitation spectrum with Hs = 0.01h and ω p = ω1 . (b) Displacement generated from the spectrum. (c) Free surface elevation at the left corner of the wall due to horizontal motions prescribed by (b). (d) Free surface elevation at the left corner of the wall due to vertical motions prescribed by (b).
and 0.288 with constant excitation amplitude are presented in Figs. 8 and 9 respectively. Such plots have been reported for ωv = 0.75ω1 , ω1 , 1.5ω1 , 2ω1 and 2.5ω1 . A comparison of the results presented in the above two figures reveals the following. In the case of lesser initial steepness, a single peak of the dimensionless sloshing energy is observed at frequency ratio of 1, irrespective of the magnitude of the excitation frequency. In the case of the higher initial steepness adopted in the study, the dimensionless sloshing frequency spectra exhibits secondary peaks at ω/ω1 = 1.5 and 2 for all the ωv tested. The magnitude of the secondary peak is insignificant compared to the primary peak at the resonance frequency. Frandsen [10] noticed a similar existence of peaks in the wave spectra. 5.2. Random wave excitation 5.2.1. General In order to understand the sloshing phenomenon, the container is subjected to random excitation under the real random sea state, with different peak frequency by keeping the total supplied energy to the system as constant. A typical input excitation spectrum and the corresponding displacement time history are given in Fig. 10(a) and (b) respectively. The horizontal and vertical container displacements are obtained from this spectrum (Eqs. (30) or (31)). The surface elevations at the left wall of the container due to horizontal (η/ h) H and vertical (η/ h)V excitation are shown in Fig. 10(c) and (d) respectively. The time step adopted is 0.06 s and the duration of the simulation is 61.44 s corresponding to 1024 data points, which is used to analyse the spectrum herein.
5.2.2. Horizontal excitation For a significant wave height of 0.006h, the tank is excited with random waves of different peak frequency (ω p = 0.35ω1 , ω p = 0.75ω1 , ω p = ω1 , ω p = 1.5 ω1 , ω p = ω3 ). The power spectra of free surface sloshing elevation at the left corner of the tank wall for various excitation peak frequencies are shown in Fig. 11. It can be seen that while the excitation peak frequency is less than the natural frequency, the sloshing spectral peaks appear only at natural frequencies (first, third and fifth mode) of the tank and no peak is visible at the excitation peak frequency as noticed in the case of regular excitation. The primary spectral peak lies at the first mode. When the excitation peak frequency is greater than the first mode of the tank system, there were some high frequency waves in the container (Fig. 11(d), (e) and (f)). Thus, in horizontal excitation, the spectral peaks appear only at the first or higher mode natural frequency, irrespective of the excitation peak frequency. A similar trend in the occurrence of spectral peaks is also noticed by Wang and Khoo [22]. The spectrum for forces and surface elevation of the tank wall was examined using FEM adopting isoparametric elements. It has been shown that the energy mainly concentrates at the natural frequency of the container and is found to dominate at the ith mode of the container when the peak frequency is close to the ith mode. 5.2.3. Vertical excitation The sloshing behavior of the tank when subjected to random vertical excitation is studied. The container was subjected to
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Fig. 11. Spectra of free surface sloshing waves at the left corner of the wall due to horizontal random excitation (Hs = 0.006h, T1 = 2π/ω1 ).
vertical excitation with the same input characteristics as that for horizontal motion for two different initial conditions. The initial conditions correspond to small amplitude with a steepness of 0.014 and large amplitude with a steepness of 0.288. The power spectra of waves at the left corner of the tank wall with the above initial conditions having steepness of 0.014 and 0.288 are shown in Figs. 12 and 13 respectively. It can be seen that inspite of the excitation frequency, the spectral peaks appear at the first modal frequency only while the initial perturbed waves are of small steepness. The spectral peaks appear at first, second and two times the first mode (parametric) frequency, if the initial perturbed waves are of large steepness, which is similar to that of the regular motion as discussed earlier for the heave mode. The dominating frequency appears at the first mode. However, the spectral peak magnitudes at the first modal frequency are of the same order, irrespective of excitation peak frequency. This phenomenon is observed irrespective of the different initial
perturbation in the tank. It should be noted that no regridding is necessary for the small steepness case, and hence the numerical damping does not play any role for the above results. However, in the case of regular motion, the magnitude of peak is higher at ωv = 2ω1 for small and high initial steepness. 6. Summary and conclusion This paper presents the behavior of the sloshing waves in a tank subjected to excitation in the horizontal and vertical directions using the inviscid flow solver. Both regular and random excitations have been considered. The salient conclusions drawn from the study are presented below. Due to regular excitation in the horizontal direction, the dominating spectral peak appears at the excitation mode, when the excitation frequency is less than the first mode, beyond which the sloshing is dominant at the natural frequency of the
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Fig. 12. Spectra of free surface sloshing waves at the left corner of the wall due to vertical random excitation for an initial steepness of 0.014 and Hs = 0.006h (T1 = 2π/ω1 ).
container. The critical sloshing motion occurs at the well known resonance condition, while the excitation frequency is equal to the first mode. Due to regular excitation in the vertical direction, the spectral peak appears to be at the first mode for small initial perturbation and at first, second and two times of the first mode for higher initial perturbation for constant excitation amplitude. The dominating spectral peaks in both the cases are at the first mode. The critical sloshing occurs at the excitation frequency equal to twice the first mode, well known as the parametric resonance. For the random excitation in the horizontal direction, the spectral peaks occur only at the natural frequencies of the container, the dominating peak appear close to ith mode, when the excitation peak frequency is at ith mode. The maximum intensity appears to be at the excitation peak frequency equal to the first mode that is, resonance condition.
In the case of random excitation in vertical direction, the dominating peak appears only at the first mode. The magnitude of the peak is almost the same irrespective of the excitation peak frequency and initial perturbation, contrary to the regular excitation, wherein the magnitude is large only if the excitation frequency is equal to twice the first mode (parametric resonance) irrespective of initial perturbation. Frandsen [10] showed that the combined effect of vertical and horizontal regular excitations lead to an infinite number of additional resonant frequency components. The numerical results of the individual excitation in horizontal and vertical directions are in good agreement with the results of Frandsen [10]. The magnitude of the peak is almost the same irrespective of the excitation peak frequency and initial perturbation for vertical random excitation. This requires further investigation in random excitation to examine the effect of the additional frequency resonance for combined random excitation.
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Fig. 13. Spectra of free surface sloshing waves at the left corner of the wall due to vertical random excitation for an initial steepness of 0.288 and Hs = 0.006h (T1 = 2π/ω1 ).
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