Seiche oscillations of layered fluids in a closed rectangular tank with wave damping mechanism

Ocean Engineering 196 (2020) 106842

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Seiche oscillations of layered fluids in a closed rectangular tank with wave damping mechanism Mi-An Xue a, b, Obai Kargbo a, b, Jinhai Zheng a, b, * a b

Key Laboratory of Coastal Disaster and Defence of Ministry of Education, Hohai University, Nanjing, 210024, China College of Harbour Coastal and Offshore Engineering, Hohai University, Nanjing, 210024, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Seiche oscillations Sloshing Layered fluids Interfacial wave Wave damping

A numerical model is employed here to study seiche oscillations of layered or stratified fluids in a closed rect­ angular tank under various initial inclination angles of the free surface and interfacial layers to the normal water table as seiche setup conditions. Laboratory experiments were also conducted for layered sloshing in a rectan­ gular tank. The numerical model was validated against the experimental data and the available numerical re­ sults. The effect of the setup angle on the maximum interfacial and free surface wave height time series were then studied in relation to their corresponding dominant frequency response. The effects of internal structures on the rate of wave decay were analyzed for internal and free surface waves with key findings being reported.

1. Introduction Seiche is free fluid oscillations synonymous to enclosed or semienclosed water bodies. In nature initiation is caused by strong wind action during storms which in some cases applies a unidirectional wind pressure load on the water free surface and this pressure pushes the lake water in the direction of the storm thereby increasing the water table on one side creating an inclined free surface. Once this storm subsides, and the applied wind pressure seizes to act, the action of gravity causes the fluid to sway in a free sloshing like manner until it reaches equilibrium. This action in extreme cases has been known to create waves which cause damage to boats, structures and even endanger the lives of humans. The multiphase study of this phenomenon considers the modeling of the fluid volume as a composite of two density stratified fluid layers as is the case in lakes which have stratification due to var­ iations water temperature, salinity content or impurities. The vertical movement caused by internal waves during this process has chemical and biological consequences in lake environments. Free oscillation or seiche has been widely investigated in water bodies like lakes, estuaries and coastal localities, thanks to their large impact on circulation flow and bio-chemical processes (Niedda and Greppi, 2007). Lemmin et al. (2005) analyzed season-long water level records at 12 stations around the Lake of Geneva for evidence of internal seiches modified by Coriolis force. Niedda and Greppi (2007) high­ lighted that the seiche oscillation is one of the mechanisms responsible

for driving residual currents in lagoon. Jordi et al. (2008) described the seiche in Mediterranean harbour as the principal mechanism that con­ trols the sediment resuspension and the variability of the suspended sediment using both observations and model simulations. Qin and Lin (2017) developed a GIS-based integrated framework for coastal seiches monitoring and forecasting system for preliminary predictions of seiche events in the North Jiangsu shoal, China. Cueva et al. (2019) presented an analysis of the natural frequencies and vibration modes of the structure of the superficial seiches of Lake Chapala using mathematical modelling and measurements. Their results provided the spatial distri­ bution of Lake water level oscillations for seiche periods. de Jong and Battjes (2004) identified the main mechanism of the generation of seiches in the Port of Rotterdam and investigated the seiche character­ istics. Chen (2005) assumed that there is no wind acting on the free surface and investigated seiche oscillation in a rectangular basin caused by a sinusoidal form initial free-surface by using a 3D nonhydrodynamic model named LESS3D (Lake and Estuary Simulation System in Three Dimensions). Ahn (2008) studied the seiche wave formation in deep and shallow water waves. Their work was able to estimate the velocity of gravity waves in the transition region between deep and shallow water waves. Bukreev et al. (2013) studied the seiche oscillation in a rectangle channel with an abrupt expansion of the cross-section. In their study, they investigated the natural frequencies of the seiche oscillation in a rectangular cross-section experimentally and numerically. Bowers et al. (2013) concluded that the tide, forcing a shallow coastal water body, can

* Corresponding author. Key Laboratory of Coastal Disaster and Defence of Ministry of Education, Hohai University, Nanjing, 210024, China. E-mail addresses: [email protected] (M.-A. Xue), [email protected] (O. Kargbo), [email protected] (J. Zheng). https://doi.org/10.1016/j.oceaneng.2019.106842 Received 30 June 2019; Received in revised form 31 October 2019; Accepted 7 December 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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produce an oscillation at the natural period of the water body namely seiche, which mean that a shallow water body (Zhang et al., 2018) ap­ pears to be a necessary requirement for the production of tide-produced seiches. Cushman-Roisin et al. (2005) studied the influence of water stratification on decaying surface seiche modes. They concluded that in enclosed water bodies such as lakes, in which external seiches are standing waves, their period is weakly dependent on the stratification; in semi-enclosed systems such as bays and coastal seas, from which surface gravity waves can radiate into the open ocean, the external seiche period is a more sensitive function of stratification. Seiche oscillation is very closely related to the sloshing phenomenon in tanks that contain a fluid with a free surface. A close look at sloshing thus gives a basis of understanding the similarities to seiche oscillations and in favor of proposing how this could be used to study seiche occurrence. Faltinsen and Timokha (2009) and Ibrahim (2015) are good sources which could broaden the readers understanding on free surface sloshing. Seibi and Perez (2007) described the setting up of a test rig to study liquid sloshing including a data acquisition system. Lin (2007) developed a model to simulate fluid sloshing in a horizontally excited tank with different excitation frequencies and the numerical results compared well with the established analytical results. Vorobyev et al. (2011) simulated centralized sloshing with a central water column collapsing within the confines of a cylindrical tank. Smoothed particle hydrodynamics (SPH) simulation on large-amplitude sloshing for fluids in a 2D tank was studied by Wang et al. (2013). In their results they deduced that the SPH method was very effective for solving nonlinear sloshing as well as for violent fluid sloshing and wave breaking. Aly et al. (2015) used a stabilized incompressible SPH method in pressure calcu­ lations in a highly nonlinear sloshing problem. Luo et al. (2016) and Luo and Koh (2017) studied the 3D sloshing in a scaled membrane-type LNG tank numerically and experimentally. Sivý et al. (2017) studied fluid structure interaction in a sloshing tank by dividing the liquid mass into impulsive and convective zones. They aimed to deal with the seismic design of open cylindrical liquid storage tanks. Xue et al. (2017a) pro­ posed a finite difference model for studying violent transient sloshing and considering turbulence. The numerical modeled matched well with an experimental case and results accounted well for strong nonlinearity. Chen and Xue (2018) studied the effect of different filling levels, exci­ tation frequency and amplitude on sloshing by using OpenFOAM and validating their model via experimental data. Jin and Lin (2019) studied the viscous effect of liquid sloshing under external excitation. They adopted a 3D two-phase flow model to study sloshing of a highly viscous fluid and validated against available analytical solutions and numerical results. Xue et al. (2019) carried out fluid dynamics analysis of sloshing pressure distribution in storage vessels of different shapes. Effects of the baffles or obstacles on free surface wave are widely reported, which can dissipate wave energy by multiple damping mechanisms (Xue and Lin, 2011). Jung et al. (2012) studied effect of vertical baffle heights on sloshing. In their study, vortexes were seen to develop in the vicinity of the baffle tip as it damps the energy in the fluid. Xue et al. (2012) conducted sloshing studies in relation to the effect of perforated baffles in a cubic tank. Their study highlighted the baffles effect on damping impact pressure on the tank walls. Koh et al. (2013) studied a novel sloshing mitigation device, i.e. the constrained floating baffle, based on the improved consistent particle method. Xue et al. (2013a) studied the effect of perforated baffles on reducing sloshing in a rectangular tank with an experimental setup and compared results with an in-house numerical wave tank model NEWTANK. Xue et al. (2017b) experimentally studied the effect of different vertical baffle configurations in reducing sloshing pressure under various exci­ tation frequencies. This study gave a detailed insight on the effectiveness

of baffles in reducing sloshing pressure. Yu et al. (2019) studied the use of a vertical slat screen in reducing in shallow water sloshing in a tank. This study showed that sloshing damping does not always increase with an increase in the screen number, but is also dependent on the position of the screen. Xiang et al. (2019) analyzed the wave clipping effects of artificial islands in plain reservoir by employing Mike21 SW model. They concluded that artificial islands at suitable locations in the reser­ voir can attenuate the wave heights by approximately 10%–30%. The study of layered sloshing can also throw light on layered seiche oscillations. A couple of studies have been conducted in the past that deal with a layered fluid sloshing in a closed tank. Kargbo et al. (2019) looked at the effect of layered sloshing and internal wave interaction on submersed structures. This study showed that internal wave progression is affected by internal structures, with interfacial waves being higher than the corresponding free surface wave. Veletsos and Shivakumar (1993) conducted studies on layered sloshing considering N layers and their study stated that for N number of homogeneous layers there are an N number of horizontal natural modes. La Rocca et al. (2002) conducted theoretical studies on layered sloshing using a variational form of the potential formulation of the fluid motion and in his study was able to account for the effects of dissipative forces. Sciortino et al. (2009) used a Hamiltonian approach to study layered sloshing in a cylindrical tank and was able to analyze the traveling waves created during sloshing. Ish­ iyama et al. (2013) looked at how a magnetic fluid affected layered sloshing. Their study discovered that the magnetic field altered the natural frequency and the pressure exerted by the fluid on the tank walls. In general, studies on seiche in lakes have been able to improve on the general understanding of seiche oscillations in nature and the de­ velopments in sloshing analyses have bolstered our understanding on this phenomenon. This work however looks at how seiche oscillations could be developed numerically for single and layered cases and how the setup affects the oscillation. How layered seiche oscillations and inter­ facial waves interact with damping structures is also an area which this study seeks to add further understanding on. 2. Mathematical model and numerical method 2.1. Governing equation The incompressible Navier-Stokes equation governs the fluid domain

ρ

∂u þ ρu ⋅ ru ∂t

r⋅

r⋅u ¼ 0

pI þ μ ru þ ðruÞT

��

¼F

(1) (2)

The velocity is given by u, with the pressure given by p, whereas the fluid density is given by ρ. The dynamic viscosity and the body force are given by μ and F respectively, with the unit diagonal matrix represented by I. 2.2. Numerical method In this study, the fluid domain of the model is represented by the incompressible Navier-Stokes equation with the ALE method being employed to track the fluid interface deformations. This model is based on the finite element method (FEM) and is developed using the CFD module of COMSOL. For the convenience of the following discussions, the time item of the momentum equation is ignored for the time being and the body force is set to zero. Therefore, the steady, laminar equa­ tions in the domain Ω with non-slip and zero normal stress conditions on the domain boundaries Γ0 and Γn respectively in component form are:

2

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�

ρuj ​ ui;j ¼ p;i þ μ ui;j þ uj;i ;j in ​ Ω; ui;j ¼ 0 in ​ Ω; ui ¼ 0 on ​ Γ0 ;

� pni ​ þ μ ui;j þ uj;i nj ¼ 0 on ​ ​ Γn ;

The non-slip boundary condition (on ​ ​ Γ0 ) is enforced explicitly on the FEM functions with appropriate selection of the function space. The zero-normal stress boundary condition (on ​ Γn ) is implicitly enforced through the variational form. P1 þ P1 discretization which apply separate discretization for the velocity and pressure is done using the continuous, piecewise linear elements for both variables. Further details on the COMSOL solver equations can be obtained from the COMSOL Users Guide (Multiphysics COMSOL, 1998) and the reference (Kargbo et al., 2019).

A test function vi is selected from the momentum equation and in such vi ¼ 0 on ​ Γ0 , where as a test function z is chosen in the case of the continuity equation. The continuity and momentum equations and are multiplied by a test function and integrated over the domain: Z Z Z Z � ρ uj ui;j vi ​ dV ¼ p;i vi ​ dV þ μ ui;j þ uj;i ;j vi ​ ​ dV ui;i z ​ dV ¼ 0 (4) Ω

Ω

Ω

Ω

Integration by parts applied to the pressure and vicious term yields: Z ​ Z ​ Z ​ p;i vi dV ¼ pni vi ​ dS þ pvi;i ​ dV (5) Ω

Z

Γn

∂Ω

Ω

Z

μ

​

� ui;j þ uj;i ​ nj vi ​ dS

Γn

2.2.1. Boundary conditions The model consists of solid walls, and a free surface. The slip boundary for the Navier-Stokes equation is

Ω

I � � ui;j þ uj;i ;i vi dV ¼ μ ui;j þ uj;i nj vi dS

Z

μ

2

Z ​

� � ui;j þ uj;i vi;j þ vj;i ​ dV

Ω

where n ¼ (nx, ny)T is the boundary normal. Equation (10) is applied to the walls which are modeled with slip boundary condition. The normal vector is dependent on the degrees of freedom of the moving mesh. Hence, weak constraints are used to remedy constraining forces on the fluid equations and mesh. Weak constraints are used to enforce the slip boundary condition without a constraint force acting on the moving mesh equations:

(6)

The sum from the boundary terms from equations (5) and (6) give zero stress boundary condition integral Z Z Z � � � � ui;j þ uj;i nj vi dS ¼ pni þ μ ui;j þ uj;i nj vi dS ¼ 0 pni vi dS þ μ Γn

Γn

b λð ​ u ⋅ nÞ

Γn

Putting expressions (5)–(7) back in the integral expression (4) gives the final variational form Z Z Z � � μ ui;j þ uj;i vi;j þ vj;i ​ dV (8) ρ ui ui;j vi ​ dV ¼ pvi;i ​ dV 2 Ω

λ ðb u ⋅ ​ nÞ ​

(11)

For some Lagrange multiplier, variable λ, where b λ and b u denote test functions. The fluid is free to move in the top boundary. Neglecting the stress in the surrounding environment, the stress continuity equation on the free surface is given by: �� pI þ μ ru þ ðruÞT ⋅ n ¼ p0 n (12)

(7)

Ω

(10)

u⋅n ¼ 0

� ui;j þ uj;i vi;j dV ¼

Ω

μ

(3)

Ω

(9)

where p0 is the surrounding pressure and μ is the viscosity of the fluid, without loss of generality, p0 ¼ 0 for this model.

For solving the unsteady flow, it is noted that the time item should be added to Equation (8). Also, all the mentioned formulations are already implanted within this software.

2.2.2. Boundary condition for mesh To follow the fluid motion in relation to the mesh motion, it is conventional in most cases to couple the mesh motion to the fluid mo­ tion normal to the surface. The coupling is not done in the tangential

Z ui;i z dV ¼ 0 Ω

Fig. 1. Test rig and diagram of the tank containing two immiscible fluids: (a) Photograph, (b) layout of the sensors. 3

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direction as it will soon lead the solution to non-convergence due to excessive deformation of the mesh. The boundary condition for the mesh on the free surface is thus given by

Table 1 Mesh characteristic analysis. Mesh Description

Max. Element size (m)

Min. Element size (m)

Max element growth rate

Curvature factor

Resolution of narrow regions

Coarse Normal Fine

0.015 0.00675 0.0021

0.15 3.0E-4 6E-5

2.5 1.15 1.1

1 0.3 0.25

0.1 1 1

ðxt ; yt ÞT ⋅ n ¼ u⋅n

(13)

where n is the boundary normal to ðxt yt )T the velocity mesh. For this boundary condition, with the weak constraints activated, the moving mesh interface adds the weak expression: � � �T � b λðððxt ; yt ÞT u ⋅n λððb χ ; by ⋅ n (14) This is added to ensure that there are no constraint forces acting on

Fig. 2. Free surface wave time series mesh error bar for fine and coarse mesh.

Fig. 3. Time step convergence test for (a) interfacial wave and (b) free surface wave. 4

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Fig. 4. Comparison of the numerical and experimental results for the time history of the interfacial wave elevations at probe 1, probe 2 and probe 3 at excitation frequency ω ¼ 4:24 ​ rad=s.

the fluid equations. The moving mesh is not constrained in the tangen­ tial direction on the side wall at the side to enable it to follow the motion of the fluid. Here again, λ denotes some Lagrange multiplier variable (not the same as before) and ​ b x; b y and b λ denote test functions.

the domain, and as such the mesh is considered as incompressible. We consider a linear elasticity model given by the constitutive relation:

σ ¼ C⋅ε

The constitutive matrix C is element dependent. The constitutive matrix elements are also inversely proportional to the element size. In order to solve the mesh equation (Eq. (15)) for free surface boundary conditions, it is assumed that no fluid particle can cross the free surface. This condition is expressed by setting the normal mesh velocity at the free surface equal to the normal fluid velocity. � v vmesh ⋅ ! n ¼0 (18)

2.2.3. Free surface and interfacial surface tracking method The free surface boundaries of the computational domain are moving and deformed with time as a result of the motion and physical param­ eters of the computational domain. The motion of this free surface is tracked by a deformed moving mesh which moves in conformity with the deformation computation domain with a new mesh being generated for each configuration of the boundaries. This technique of movement which is also referred to as the Arbitrary Lagrangian-Eulerian (ALE) method is used to track the free surface of this model. Consider a mesh at time tn, the velocity field is computed for the time tnþ1 on this same mesh. The new free surface is then computed with the Lagrangian formula. x

nþ1

n

¼ x þ uðΔtÞ

The fluid velocity is given by v is and ! n is the unit normal vector. The above-mentioned free surface tracking method is also employed to track the interfacial surface. 3. Model validation

(15)

3.1. Mesh convergence

The partial differential equation governing the displacement field of the mesh is given by: divðσ Þ ¼ 0

(17)

Mesh convergence analysis was done for the tank setup described in Fig. 1. The mesh parameters were categorized into fine, normal and coarse with the specifications of each mesh as described by Table 1. The movement equation of the tank is xðtÞ ¼ acosωt. With excitation

(16)

The Cauchy stress tensor is σ. The mass of the fluid is considered in 5

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Fig. 5. Comparison of the numerical and experimental results for the time history of the interfacial wave elevations at probe 1, probe 2 and probe 3 at excitation frequency ω ¼ 5.15 rad/s.

amplitude of a ¼ 0.01m and frequency of ω ¼ 5.15 rad/s. The mesh categories described in Table 1 were used to conduct error analysis. The normal mesh was used a reference mesh and compared to the coarse and fine mesh by plotting error bars as shown in Fig. 2 for coarse mesh error and fine mesh error respectively. From the convergence plots, it is seen that the fine mesh and the normal mesh had a strong similarity. The coarse mesh also had satisfactory comparisons with the normal mesh. For computational time-saving purposes the normal mesh was chosen throughout this study. Time step analysis was also done to see the model’s response to various time steps. Time step values of time step used were 0.1s, 0.01s and 0.001s. A rectangular tank described in Fig. 1 was used. The movement equation of the tank is xðtÞ ¼ acosωt with amplitude of a ¼ 0.01m and frequency of ω ¼ 5.15 rad/s. From the results show in Fig. 3 (a) and Fig. 3 (b) for interfacial and free surface respectively, it is seen that for the time steps used the difference in result was very minimal. For computational purposes, a time step of 0.01s was used.

ments of two-layers sloshing conducted at State Key Laboratory of Hy­ draulics and Mountain River Engineering, Sichuan University (Xue et al., 2013b). The test system including experimental photograph and layout of the sensors is shown in Fig. 1. The details are also found in previous work (Xue and Lin, 2011). The test system consisted of a layered fluid in a rectangular tank, an irregular wave-maker, three wave gauges, three pressure sensors, one displacement sensor and data acquisition system. The tank motion is controlled by an irregular wave-maker. The tank with length L ¼ 0.57m was filled with two immiscible fluids 0# diesel oil and tap water, their densities are ρ1 ¼ 846:2 ​ kg=m3 and ρ2 ¼ 1000 ​ kg=m3 respectively. The height of two layers fluids with density ρ1 and ρ2 are Hρ1 ¼ 0:05m and Hρ2 ¼ 0:1m respectively. Three wave gauges and three pressure sensors were mounted on the rectangular tank, as shown in Fig. 1(b). The movement function of the shake table controlled by the irregular wave-maker is given by:

3.2. Layered sloshing in a rectangular tank

where a is 0.01m with various excitation frequencies being ω ¼ 4:24 ​ rad=s, ω ¼ 5:15 ​ rad=s.

xðtÞ ¼

The numerical model was further compared to laboratory experi­ 6

acosωt

(19)

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Fig. 6. Comparison of the numerical and experimental results for the time history of the dynamic pressure at the pressure sensors P1,P2 and P3 at excitation frequency ω ¼ 4.24 rad/s.

The comparison of the layered sloshing in a rectangular tank of the experimental results to the numerical simulation is shown in Figs. 4–5 for the interfacial wave elevation time series and in Fig. 6 for the pres­ sure time series. As shown in Figs. 4 and 5, the results from the Probe 1 were under-predicted by the numerical model. The possible reason is that in the experiments the wave Probe 1 is not strictly in the center of the tank, where the wave amplitude is very small and the wave ampli­ tude is sensitive to position. Fortunately, the numerical results from the wave Probes near the tank walls (Probe 2 and Probe 3) and the pressure sensors match well with the experimental data. Moreover, the experi­ mental snapshots and numerical profiles of the layered profile at various moments are presented in Fig. 7. It is noted that the higher frequency interfacial waves were not captured numerically due to neglecting the effect of surface tension at the interfacial surface in numerical model. But nonetheless the agreement between the numerical results and experimental data was also satisfactory, indicating that the model employed here is a robust tool for simulating layered fluids sloshing problem.

sloshing in rectangular tanks partially filled with liquid. The tank has a length L of 0.9m and a liquid filling level H of 0.6m. In case 1, the height of the bottom mounted baffle is 0.3m and is placed at the bottom center of the tank and in case 2 the height of the vertical baffle is 0.075m and is placed near the free surface as is seen in Fig. 8. The tank motion is subjected to the following sinusoidal function; xðtÞ ¼ a sin ω t

(20)

where the amplitude is a ¼ 0.00198m, and the excitation frequency is

ω ​ ¼ 5.4 rad/s. Fig. 9 shows the present results of the free surface

elevation time history at the left wall compared well with the available numerical results from Belakroum et al. (2010). However, slight discrepancy in amplitude and slight phase lag is observed in the com­ parisons due to considering the impulse acceleration in present simu­ lations. Overall, the present results however are in a satisfactory agreement with the available numerical results. Moreover, Fig. 9 (a) and (b) indicates the temporal evolution of the free surface elevations at the left wall after installing the vertical baffle placed at the bottom of the tank and near the free surface respectively. It can be observed obviously that the vertical baffle near free surface is more effective in reducing sloshing amplitude than the vertical baffle mounted on the bottom of the tank although its length is only one quarter of the vertical baffle on the bottom.

3.3. Liquid sloshing in a tank with vertical baffles A vertical baffled tank model was setup as shown in Fig. 8 and compared for validation with numerical results from Belakroum et al. (2010), who investigated effect of the baffles on passive reducing 7

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Fig. 7. Comparison of the numerical and experimental results for the free surface profile and the interfacial profile at different moments at excitation frequency ω ¼ 4.24 rad/s.

4. Single layer induced seiche oscillations in a rectangular tank 4.1. Model setup To replicate the seiche setup phenomenon and action as observed in lakes and enclosed water bodies, a case is setup with an initially inclines free surface as shown in Fig. 10 with a closed tank of dimensions and fluid domain surface area as described in Table 2. No motion excitation amplitude and frequency were applied to the system to initiate sloshing, but the fluid motion is caused by gravity acting on the initially inclined fluid surface on the commencement of the simulation. For the assump­ tion of a flat bottom, the characteristic period T of a barotropic seiche is given by .� pffiffiffiffiffi� T ¼ 2L m gh (21) where m ¼ 1, 2, 3, …; L is the length of the water basin; h¼(H1þH2)/2 is the water depth; and g is the gravitational acceleration (Niedda and Greppi, 2007). Four case studies are modeled as shown in Table 2, namely Case 1, Case 2, Case 3 and Case 4 respectively. As is seen in Table 2, the volume of the fluid is kept constant for all cases with changes only being made to the inclination angle of the free surface. The area of the trapezium shown in Table 2 is considered for the 2D setup of the fluid domain.

Fig. 8. Two-dimensional rectangular tanks with different vertical baffles. 8

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Fig. 9. Comparison of present numerical results (solid line) compared with results obtained from Belakroum et al. (2010) (dotted line) of relative elevation time series for vertical baffle placed at: (a) the bottom of the tank; (b) near the free surface.

4.2. Wave time series and spectral analysis for single seiche oscillations As shown in Fig. 10, free surface wave heights were recorded on the right and left tank walls as C1 and C2 respectively. Seiche setup was modeled with different free surface inclination angles, and at the commencement of the simulation the fluid mass can oscillate under gravity till it reaches equilibrium, as observed in the free surface time series shown in Figs. 11–14. The inclination angle was developed using the details and dimensions of the four cases studied as given in Table 2. To alter the inclination angle of the free surface, the height of H2 in the tank setup was varied giving free surface inclinations related to H2 ¼ 0.115m, 0.125m, 0.145m and 0.165m, with their respective sloshing wave time series being plotted in Figs. 11–14 respectively as well as their respective fast Fourier transform (FFT). A FFT provides a power spectral analysis technique of the time series and enables the natural or peak frequencies of the system to be identified. From the FFT diagrams, high frequency components at high inclined fluid surface conditions can be observed. This may be due to the rapid descending of the inclined fluid surface under gravity, which may cause fast oscillation of fluid particles in the process. Meanwhile, a typical nonlinear wave phenomenon, i.e. the wave crest becomes sharper and the trough becomes flatter with larger inclination angle is obviously observed in Fig. 15, indicating that wave-wave interactions are enhanced by increasing inclination angle. More high frequency components are thus observed in Figs. 12–14. The

Fig. 10. Initial fluid domain in closed tank with inclined free surface.

Table 2 Computational fluid domain initial parameters. Parameters

Case1

Case2

Case3

Case4

H1 H2 L Area (trapezium) θ

0.101m 0.115m 0.500m 0.054m2 91.60�

0.091m 0.125m 0.500m 0.054m2 93.89�

0.071m 0.145m 0.500m 0.054m2 98.42�

0.051m 0.165m 0.500m 0.054m2 102.84�

Fig. 11. Seiche initiated sloshing free surface time series wave for H2 ¼ 0.115m and corresponding FFT.

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Fig. 12. Seiche initiated sloshing free surface time series wave for H2 ¼ 0.125m and corresponding FFT.

Fig. 13. Seiche initiated sloshing free surface time series wave for H2 ¼ 0.145m and corresponding FFT.

Fig. 14. Seiche initiated sloshing free surface time series wave for H2 ¼ 0.165m and corresponding FFT.

Fig. 15. Seiche initiated sloshing free surface time series wave for various H2 values.

free surface time series showed that at the initial commencement of the simulation, the fluid sloshed from side to side with wave heights decreasing as the time progresses until equilibrium is obtained. Figs. 11–14 shown that the higher H2 values the sloshing fluid had higher wave heights and took longer to reach equilibrium. This can be

seen obviously in Fig. 14, where H2 ¼ 0.165m and the free surface wave time series is seen to have higher wave heights as compared to the other cases studied in Figs. 11–13. Lowest wave heights for all the cases studied is observed when H2 ¼ 0.115m as shown in Fig. 11. A comparative analysis shown in Fig. 15 compares all cases for values of 10

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Fig. 16. Velocity profile for seiche initiated sloshing under various initial inclination angle setup.

Fig. 17. Initial fluid domain in a closed tank with inclined free surface and interfacial layer for a baffled and unbaffled case.

11

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Fig. 18. Seiche initiated sloshing and free surface and interfacial layer wave time series for H2 ¼ 0.165m and corresponding FFT for closed tank with and without baffles.

H2 ¼ 0.115m, 0.125m, 0.145m and 0.165 m at right and left tank walls by wave gauges C1 and C2 respectively. This further shows that at higher values of H2 or inclination angle is associated with higher wave heights and longer sloshing time of the free surface before it reaches equilibrium. Power spectral analysis for single layer seiche under setup as described by Fig. 10 and Table 2 is plotted for various initial free surface inclination angles for H2 ¼ 0.115m, 0.125m, 0.145m and 0.165m as shown in Figs. 11–14. The frequency analysis was done for wave heights at C1 and C2 and it is observed that the peak frequency values at C1 and C2 were similar in value. This can be seen in Fig. 11 where the peak frequency is given as 0.93 Hz, which is around the first-order natural frequency calculated by equation (21). For H2 ¼ 0.125m the peak fre­ quency is also given as 0.93 Hz as seen in Fig. 12. As H2 is increased to H2 ¼ 0.145m as shown in Fig. 13, thereby increasing the free surface inclination angle, the peak frequency is given as 0.96 Hz which showed a slight increase from the previous H2 values of Figs. 11 and 12. It is also observed that for H2 ¼ 0.165m the peak value is around 0.96 Hz which is very close to the peak frequency when H2 ¼ 0.145m. From the spectral analysis it can be summed that for the same filling levels or for a certain volume of fluid, the natural or peak frequency is not significantly altered by varying the inclination angle of the free surface.

4.3. Comparison of velocity profile of seiche oscillations under various initial free surface inclination angle Velocity screen shots were taken for seiche initiated sloshing under four different initial free surface inclination angles. The free surface inclination angles were given by H2 ¼ 0.115m, 0.125m, 0.145 m and 0.165m as is seen in Fig. 16. From the screen shots the fluid is seen to slosh from side to side for all H2 heights. The velocity distribution varied for all inclination angles with one notable observation being that for higher H2 values a higher maximum velocity of the fluid domain is observed and this can clearly be seen at t ¼ 1.3s. This showed that an increase in inclination angle caused an increase in the fluid velocity as it sloshes from side to side. Interesting velocity distributions can be observed at certain time frames. At t ¼ 2.77s, it is seen that the screen shot for H2 ¼ 0.115m has a velocity distribution characteristic of central larger and surrounding smaller with compared to the other cases of H2 ¼ 0.125m, 0.145m and 0.165m, i.e. the fluid velocity is concentrated in the center, with higher velocity in the center being encircle by lower velocity distribution, whereas the maximum velocity concentration is closer spread out and closer to the free surface for the other values of H2. Similar interesting velocity differences can be observed at t ¼ 5.79s where for all values of H2 each had a uniquely different velocity dis­ tribution profile at this time frame.

Fig. 19. Seiche initiated sloshing free surface and internal layer wave time series for H2 ¼ 0.145m and corresponding FFT for closed tank with and without baffles. 12

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Ocean Engineering 196 (2020) 106842

Fig. 20. Seiche initiated sloshing free surface and internal layer wave time series for H2 ¼ 0.125m and corresponding FFT for closed tank with and without baffles.

Fig. 21. Wavelet analysis of time series of free surface and interfacial wave under seiche action for initial fluid surface inclination with H2 ¼ 0.125m.

5. Two-layered induced seiche oscillations in a rectangular tank

spectral analysis. It is seen that the baffles had an effect in reducing the interfacial and free surface wave displacement height as well as bringing the sloshing fluid to attain equilibrium in a shorter time, thereby speeding up the creation of a still fluid surface. This can be seen in Fig. 18 which shows seiche sloshing for the interfacial and free surface layer for H2 ¼ 0.165m for cases with and without a baffle. In Fig. 18, the interfacial and free surface wave interacting with a baffle is seen to quickly reach equilibrium with lesser sloshing heights as compared to the same case without a baffle. This can be further observed in Fig. 24 which shows screen shots of the velocity profile of the seiche for various initial inclination angles. This is also seen in for other initial inclination angles of the free surface. The baffled cases can also be seen to have significant second peak frequencies values with high amplitudes as compared to the unbaffled case where in the first peak frequency is significantly higher than the second peak frequency. This can be seen in the FFT of Figs. 18–20. It is seen that the first sloshing cycle attains the highest maximum wave height with subsequent wave heights reducing gradually till the initial and free surface attain equilibrium with a still fluid table. As the initial inclination angle of the free surface is reduced,

5.1. Model setup A setup was developed to study multiphase or layered seiche comprising of two immiscible fluids in a closed tank as is seen in Fig. 17. The tank and domain dimensions are as described by Table 2. Two cases were setup for studies with an inclined layered fluid domain for an unbaffled and central-mounted vertical baffles case as in described in Fig. 17. This analysis is done for cases with and without a baffle using the cases as describe in Table 2, with the four cases shown in the table being developed with a second fluid (Layer 2) having a height of H3 being placed on top and parallel to the bottom fluid as shown in Fig. 17. 5.2. Wave time series and spectral analysis for two-layered seiche oscillations Time series graphs are shown for the interfacial and free surface evolution for layered seiche, together with their corresponding power 13

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Fig. 22. Wavelet analysis of time series of free surface and interfacial wave under seiche action for initial fluid surface inclination with H2 ¼ 0.145m.

Fig. 23. Wavelet analysis of time series of free surface and interfacial wave under seiche action for initial fluid surface inclination with H2 ¼ 0.165m.

the fluid attains lower maximum wave heights and reaches equilibrium much faster as is seen in Fig. 20 for H2 ¼ 0.125m. Corresponding power spectral analysis of the interfacial and free surface wave time series are presented for both baffled and unbaffled cases. In Fig. 18 for H2 ¼ 0.165m, it is seen that the baffle split the peak frequency without baffle into two dominant response frequencies. In Fig. 18 for the internal or interfacial wave, the peak frequency is given as 0.99 Hz for unbaffled case of the interfacial waves and 0.92 Hz and 1.64

Hz when a baffle is used. It is also seen that the third peak frequencies have very small amplitudes when compared to the peak frequencies when considering the unbaffled cases. Similarly, for the free surface, the baffle split the same peak frequency of 0.99 Hz into two dominant fre­ quencies 0.88 Hz and 1.60 Hz. This can be also observed in Figs. 19 and 20 for the cases without and with the baffle under various initial free surface inclination angles. Moreover, it was observed that effect of the baffle on frequency domain of the free surface elevation time series was 14

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Ocean Engineering 196 (2020) 106842

Fig. 24. Velocity profile for seiche initiated sloshing under inclination angle setup of H2 ¼ 0.165m for unbaffled and baffled tank.

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different from that of the internal wave elevation. The spectral density analysis mainly shows the energy distribution in frequency domain. This also shows a relation between the inclination angle and the two layer fluids energy distribution in frequency domain. However, the fluids energy distribution in time was not observed by using FFT method. Wavelet transform, which is a time-frequency analysis method, can provide local information and therefore may be viable to identify the time-frequency characteristics of the time series of free surface and interfacial wave (Ma et al., 2010; Tai et al., 2019). In Figs. 21–23, wavelet spectrum are shown for the time series for H2 ¼ 0.125m, 0.145m and 0.165m respectively, which obviously exhibits the energy distribution of external wave and internal wave in different frequencies and its variation over the time.

observed at special time. Declaration of competing interest The authors declare no conflict of interest. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 51679079), the Fundamental Research Funds for the Central Universities, China (No. 2018B12814), the National Science Found for Distinguished Young Scholars (No. 51425901), and the Pro­ gram for Excellent Innovative Talents of Hohai University, China.

5.3. Effect of the baffles on velocity profile of seiche oscillations

Appendix A. Supplementary data

The seiche oscillations for H2 ¼ 0.165m for cases with and without a vertical baffle were compared with results of the velocity screen shots shown in Fig. 24. The effect of the baffle on the oscillation was desired to be observed for the highest H2 value covered in this study to observe how high inclination angle of the free and interfacial surfaces interact with and without internal structures such as baffles. The velocity dis­ tribution obviously varied for the case with a baffle when compared to that for the unbaffled case. For the baffled case obvious vortex shedding can be observed near the tip of the baffle which in turn might dissipate fluid energy as opposed to the case without a baffle. This can be clearly observed in Fig. 24 at t ¼ 1.62 s. This same observation can be deduced for the other time frames of Fig. 24. At t ¼ 1.62 s and 2.77 s, it can be observed that the baffled case created interfacial wave high frequency oscillation and an upward interfacial wave displacement just above the baffle position. This displacement is not observed for the unbaffled case. The baffle here is seen to interrupt with the hydrodynamic flow path of the fluid with particular effects been observed on the interfacial wave.

Supplementary data to this article can be found online at https://doi. org/10.1016/j.oceaneng.2019.106842. References Ahn, D., 2008. Seiche wave formation in deep-water and shallow-water waves. ISB J. Phys. 2 (2), 1–4. Aly, A.M., Nguyen, M.T., Lee, S.W., 2015. Numerical analysis of liquid sloshing using the incompressible smoothed particle hydrodynamics method. Adv. Mech. Eng. 7 (2), 1–14. Belakroum, R., Kadja, M., Mai, T.H., Maalouf, C., 2010. An efficient passive technique for reducing sloshing in rectangular tanks partially filled with liquid. Mech. Res. Commun. 37 (3), 341–346. Bowers, D.G., Macdonald, R.G., McKee, D., Nimmo-Smith, W.A.M., Graham, G.W., 2013. On the formation of tide-produced seiches and double high waters in coastal seas. Estuar. Coast Shelf Sci. 134, 108–116. Bukreev, V.I., Sturova, I.V., Chebotnikov, A.V., 2013. Seiche oscillations in a rectangular channel. J. Appl. Mech. Tech. Phys. 54 (4), 531–540. Chen, X.J., 2005. A comparison of hydrostatic and nonhydrostatic pressure components in seiche oscillations. Math. Comput. Model. 41, 887–902. Chen, Y.C., Xue, M.-A., 2018. Numerical simulation of liquid sloshing with different filling levels using OpenFOAM and experimental validation. Water 10 (12), 1–18. Cushman-Roisin, B., Willmott, A.J., Biggs, N.R.T., 2005. Influence of stratification on decaying surface seiche modes. Cont. Shelf Res. 25, 227–242. Cueva, D.A., Monz� on, C.O., Filonov, A., Tereshchenko, I., Covarrubias, P.L., Gonz� alez, J. R.G., 2019. Natural frequencies of seiches in lake Chapala. Sci. Rep. 9, 11863. de Jong, M.P.C., Battjes, J.A., 2004. Seciche characteristics of Rotterdam harbour. Coast. Eng. 51, 373–386. Faltinsen, O.M., Timokha, A.N., 2009. Sloshing, first ed. Cambridge University Press, Cambridge, pp. 1–606. Ibrahim, R.A., 2015. Recent advances in physics of fluid parametric sloshing and related problems. J. Fluid. Eng. - T. ASME 137, 1–52. Ishiyama, T., Kaneko, S., Takemoto, S., Sawada, T., 2013. Dynamic pressure change of two-layer sloshing using a magnetic fluid. In: 8th Symposium on Flow Manipulation and Active Control: Theory, Experiments and Implementation, 1B, pp. 1–7. Jin, X., Lin, P.Z., 2019. Viscous effects on liquid sloshing under external excitations. Ocean Eng. 171 (1), 695–707. Jordi, A., Basterretxea, G., Casas, B., Angl� es, S., Garc�es, E., 2008. Seiche-forced resuspension events in a Mediterranean harbour. Cont. Shelf Res. 28 (4–5), 505–515. Jung, J.H., Yoon, H.S., Lee, C.H., Shin, S.C., 2012. Effect of the vertical baffle height on the Liquid sloshing in a three-dimensional rectangular tank. Ocean Eng. 44, 79–89. Kargbo, O., Xue, M.-A., Zheng, J.H., 2019. Multiphase sloshing and interfacial wave interaction with a baffle and a submersed block. J. Fluid. Eng. - T. ASME 141, 1–15. Koh, C.G., Luo, M., Gao, M., Bai, W., 2013. Modelling of liquid sloshing with constrained floating Baffle. Comput. Struct. 122, 270–279. La Rocca, M., Sciortino, G., Boniforti, M.A., 2002. Interfacial gravity waves in a two-fluid system. Fluid Dyn. Res. 30 (1), 31–63. Lemmin, U., Mortimer, C.H., B€ auerle, E., 2005. Internal seiche dynamics in lake Geneva. Limnol. Oceanogr. 50 (1), 207–216. Lin, P.Z., 2007. A fixed-grid model for simulation of a moving body in free surface flows. Comput. Fluids 36 (3), 549–561. Luo, M., Koh, C.G., Bai, W., 2016. A three-dimensional particle method for violent sloshing under regular and irregular excitations. Ocean. Eng. 120, 52–63. Luo, M., Koh, C.G., 2017. Shared-Memory parallelization of consistent particle method for violent wave impact problems. Appl. Ocean Res. 69, 87–99. Ma, Y., Dong, G., Ma, X., Wang, G., 2010. A new method for separation of 2D incident and reflected waves by the Morlet wavelet transform. Coast. Eng. 57 (6), 597–603. Multiphysics COMSOL, 1998. Introduction to COMSOL Multiphysics®. accesed Nov, 27, p. 2018. https://cdn.-comsol.com/documentation/5.3.1.229/Introduction To COMSOL Multiphysics.pdf. Niedda, M., Greppi, M., 2007. Tidal, seiche and wind dynamics in a small lagoon in the Mediterranean Sea. Estuar. Coast Shelf Sci. 74, 21–30.

6. Conclusions An incompressible Navier-Stokes solver for simulating seiche phe­ nomena was built based on the CFD module of COMSOL with the interface being tracked by the Arbitrary-Lagrangian-Eulerian (ALE) method. Laboratory experiments of layered sloshing in a rectangular tank were conducted to validate the present numerical model. Good agreements are obtained for the interfacial wave elevations and the pressures between the numerical results and the experimental data. Non-layered and layered seiche was simulated by using the present numerical model. Time series plots of the interfacial and free surface wave were plotted as well as their corresponding FFT to identify their dominant response frequency of the seiche oscillation system. It showed that conditions of the wave setup influence the maximum elevation of the free surface and interfacial wave. Higher setup in inclination angles or heights created higher maximum wave height oscillations as well as longer wave decay time. The dominant response frequency values observed through the FFT were not greatly affected by the inclination angle, with peak frequency values being mostly the same for all incli­ nation angles for the constant fluid volume used in this study, which were also around the natural frequencies calculated theoretically. The maximum wave height was dependent on the inclination angle with lower inclination angle recording lower maximum wave height values. Higher velocity values were observed in the fluid domain for higher initial inclination angle values or setup height at any given time. The effect of internal baffles on layered seiche oscillation was also studied, with the internal baffles seen to greatly reduce the wave heights for both surface and interfacial waves. The baffles accelerated the wave decay for both free surface wave and interfacial wave and also affected the dominant response frequencies of the seiche oscillations. The influence of the baffle on interfacial wave and free surface wave is different, interfacial wave high frequency oscillation near the baffle tip was 16

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