A dynamic adaptation method based on unstructured mesh for solving

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A dynamic adaptation method based on unstructured mesh for solving sloshing problems Article in Ocean Engineering · December 2016 DOI: 10.1016/j.oceaneng.2016.11.016

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Ocean Engineering 129 (2017) 203–216

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A dynamic adaptation method based on unstructured mesh for solving sloshing problems Muhammad Sufyan, Long Cu Ngo, Hyoung Gwon Choi

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Department of Mechanical and Automotive Engineering, Seoul National University of Science and Technology, 323 Gongneung-ro, Nowon-gu, Seoul 139743, Republic of Korea

A R T I C L E I N F O

A BS T RAC T

Keywords: Dynamic adaptation Unstructured mesh Local coarsening/refinement Sloshing Finite element method

A dynamic mesh-adaptation algorithm based on an unstructured mesh was used to solve sloshing problems by a finite element method. The free surface evolution of sloshing with a low filling ratio is so violent that a local mesh coarsening/refinement (LCR) technique can be effectively used to resolve the flow field near the interface of two immiscible fluids. Since an implicit discretization method was employed to solve the incompressible Navier-Stokes equations, an assembled global matrix was generated using a dynamic compressed sparse row (CSR) method. The proposed algorithm was validated by comparing the simulation results with those of a nonadaptive method with respect to wall impact pressures and mass conservation. Numerical results showed that relative mass error strongly depends on mesh resolution near the interface. Results of adaptive simulations were found to be comparable with those of non-adaptive simulations only if a similar mesh resolution was used near the interface. Moreover, adaptive simulations were about two times faster than the non-adaptive ones. The effect of the adaptive zone and smoothing zone on the impact pressure was also examined for the proposed algorithm.

1. Introduction Sloshing in hydrodynamics can be described as free-surface flow in a container that is subjected to a forced motion. Sloshing problems in offshore structures and ships are of great concern to naval architects, ocean engineers, and design engineers as they are directly related to the stability of ships and offshore structures. Increasing demand for largersize tankers and liquefied natural gas (LNG) carriers requires more attention to the accurate prediction of sloshing fluid behavior in tanks. Sloshing problems are also found in other important engineering areas, such as production processes, fuel tanks on spacecraft (Hall et al., 2015), tuned liquid dampers (Molin and Remy, 2013), cylindrical cargo tanks (Hasheminejad et al., 2014), and liquid tanks on highway trucks and railroad cars. 1.1. Numerical methods in sloshing In early sloshing studies carried out in 1960s, several analytic methods were proposed to investigate the phenomenon. Later, (Faltinsen, 1974) proposed a non-linear analytical method using a perturbation technique applied to a potential flow formulation and then many variations of this approach followed. These methods were

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used to investigate simple and mild sloshing problems and provided the fundamental understanding of sloshing. However, liquid sloshing behavior is highly nonlinear, accompanied by violent motion of free surface. Analytical methods have clear limitations for these practical problems. Therefore, sloshing analysis requires numerical methods that can handle unsteady, non-linear free-surface flows. Many researchers have solved the Laplace equation based on potential flow theory to analyze sloshing with different numerical models, such asboundary-element methods (Faltinsen, 1978; Nakayama and Washizu, 1981), finite-element methods (FEM) (Cho and Lee, 2004; Nakayama and Washizu, 1980; Wang and Khoo, 2005; Wu et al., 1998), and finitedifference methods (Frandsen and Borthwick, 2003; Frandsen, 2004). However, they were unable to capture viscous flow and the rotational motion of the liquid because those studies were based on the potential flow assumption. To take into account the viscous stresses in sloshing, (Armenio and La Rocca, 1996) employed a finite-difference method for solving the incompressible Navier-Stokes equations. They compared the results of the Navier-Stokes equations to those from shallow water equations (SWE). They reported that the Navier-Stokes equations provide more accurate results than the SWE model. Later, other researchers (Chen and Chiang, 1999; Chen and Nokes, 2005; Chen, 2005) used Navier-

Corresponding author. E-mail address: [email protected] (H.G. Choi).

http://dx.doi.org/10.1016/j.oceaneng.2016.11.016 Received 26 November 2015; Received in revised form 20 October 2016; Accepted 17 November 2016 0029-8018/ © 2016 Elsevier Ltd. All rights reserved.

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narrow interfacial band. A level-set function is used as the criterion for adaptation of an unstructured grid. All previous studies of sloshing based on the level set method employed a PDE-based approach for reinitialization of the level-set function (Bai et al., 2015; Chen et al., 2009; Wang et al., 2011). In this paper, we implemented a recent scheme of re-initialization (Ngo and Choi, 2015) named “direct redistance”. Details of this scheme are presented in Section 2.2.2. Sloshing at low filling conditions is hard to simulate due to severe changes in the free surface shape and the traveling wave. (Rhee, 2005) studied sloshing at a low filling level and reported the existence of a traveling wave. (Li et al., 2014) reported that at lower filling levels, higher peak pressures were observed over the entire range of tank excitation considered in the study. Hence, accurate simulation at low filling conditions is considered as a rigorous validation of the proposed method. Section 2 of the present paper provides a detailed description of the local adaptive mesh refinement approach and the numerical method used in this study. Section 3 presents the numerical results and discussion. Section 4 presents a summary and conclusion.

Stokes equations to analyze sloshing using coordinate transformation. Meanwhile, (Kim et al., 2004) employed a solution algorithm (SOLA) scheme and adopted the concept of a buffer zone to calculate the impact pressure on the tank ceiling. Results were found to be in good agreement with other numerical results and experimental data. They used a height function to track the free surface, which restricted the free surface to be single-valued. Thus, their model is not suitable for simulating highly nonlinear sloshing with breaking and merging of the free-surface. (Marsh et al., 2011) employed a smooth-particle hydrodynamics (SPH) method to investigate sloshing; their results were in good agreement with literature. To study sloshing by solving the Navier-Stokes equations, one needs an additional scheme to capture the free-surface motion. Generally, an interface tracking method and a front capturing method are used to trace the position and shape of the free surface. The interface capturing methods are most promising for sloshing analysis, because these methods can handle the topological changes of free surface easily and are more economical. The volume-of-fluid (VOF) method is a popular interface capturing method and has been widely applied to sloshing studies (Akyildiz and Ünal, 2006; Celebi and Akyildiz, 2002; Elahi et al., 2015; Lee et al., 2007; Liu and Lin, 2008; Rhee, 2005). Recently, the level set method has also gained popularity in studying free-surface flows (Lin et al., ,2005; Yue et al., 2003) and sloshing (Bai et al., 2015; Chen et al., 2009; Jung et al., 2015; Wang et al., 2011; Zhang et al., 2014) because of its ability to easily handle the complex free-surface.

2. Numerical method In this section, the numerical method employed for the simulation of sloshing problems is described in detail. A four-step splitting FEM is explained for the solution of the incompressible Navier-Stokes equations and a level-set method combined with direct redistance for capturing the interface of sloshing problems is briefly summarized.

1.2. Adaptive mesh refinement for free-surface flows

2.1. A fractional 4-step method for the incompressible Navier-Stokes equation

Adaptive mesh refinement (AMR) can facilitate high-resolution simulations of complicated flow with greater accuracy than fixed-grid methods, without making excessive demands on computational resources. AMR typically uses one of two basic strategies: (i) re-meshing, or (ii) local refinement and coarsening. In the remeshing process, the computational grid evolves and/or deforms so as to cluster the grid points in specific regions where high resolution is deemed necessary. (Liao et al., 2000) used a remeshing scheme with an LS method. In LRC methods, a new mesh is created directly in the physical space by adding or removing computational elements to achieve a desired level of accuracy. The level of accuracy can be determined according to a number of criteria such as a posteriori error estimates and the maximum variation of a quantity across a mesh element. LRC algorithms are efficient because they do not reconstruct the whole mesh. Moreover, interpolation is cheaper in the LRC approach than in remeshing. LRC algorithms have been used in the volume-of-fluid method (Ginzburg and Wittum, 2001; Li et al., 2014; Ubbink and Issa, 1999; Wang et al., 2004) and the level-set method (Nourgaliev et al., 2005; Sochnikov and Efrima, 2003; Wang and Xiang, 2013). (Wang et al., 2004) employed a finite-volume VOF method for simulating free surface flow on dynamically adaptive quad-tree grids. The method was validated for wave motion in a two-dimensional tank. Results were in good agreement with the results of (Wu et al., 2001), who derived an analytical solution of the linearized Navier-Stokes equations for a similar configuration. To our knowledge, adaptive mesh refinement is rarely used in sloshing. (Li et al., 2014) employed a VOF method coupled with the Navier-Stokes equations to study two-dimensional sloshing by using a tree-based adaptive solver with a structured grid. Free-surface motion in a rolling square tank was investigated by numerical and experimental approaches.

A sloshing problem is defined as the motion of liquid inside a tank that is subject to periodic motion. In the present work, we employed a fractional 4-step method based on P1P1 FEM (Choi et al., 1997) to solve the incompressible Navier-Stokes equations. In the P1P1 FEM, an identical linear basis function is employed for both velocity and pressure variables. An Arbitrary Lagrangian Eulerian (ALE) method was used to account for the periodic motion of the sloshing tank. Then, the splitting formulation based on the P1P1 FEM is written as follows:

uˆi − uin 1 1 1 + ((uˆj − ug ) uˆi, j + (ujn − ug ) uin, j ) = − p n + (σˆij + σijn ), j Δt 2 ρ (ϕ ) , i 2 + Sin,

(1)

ui* − uˆi 1 = p n, ∆t ρ (ϕ ) , i

(2)

⎞ ⎛ 1 1 p n +1⎟ = ui*, i , ⎜ ⎝ ρ (ϕ) , j ⎠, j ∆t

(3)

uin +1 − ui* 1 =− p n +1 ∆t ρ (ϕ ) , i

(4)

where Δt is the time step, uˆ ,u* and ug are the two intermediate velocities and the translation velocity of the sloshing tank, superscript n denotes the time level, ρ (ϕ) is the density, S is body force , σij =ν (ui, j +uj, i ) is the viscous stress and ν = μ (ϕ)/ ρ (ϕ) is the kinematic viscosity; ϕ is the LS function defined in the next paragraph. The convection and diffusion terms are integrated using the Crank-Nicolson scheme (second-order accuracy). The pressure gradient term is decoupled from those for convection, diffusion and other external forces. In this procedure, the intermediate velocities do not necessarily satisfy the continuity equation. In the next step, the pressure is obtained from the continuity constraint and then the velocity is corrected using the pressure. For more details, see reference (Choi et al., 1997).

1.3. Contribution of present work As discussed in Section 1.2, sloshing simulations based on adaptive mesh refinement were conducted by using a VOF method with structured grids. In the present study, we propose an efficient local mesh refinement/coarsening technique that refines the mesh inside the 204

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2.2. A level set method for multiphase flow simulation 2.2.1. A least-square finite element method for the advection equation of the level-set method In the LSM, the free-surface is defined by the zero level set of an LS function (ϕ)which is advected by solving the level set transport equation with a given velocity field u:

∂ϕ +u∙∇ϕ=0 ∂t

(5)

The free surface motion is implicitly represented by the propagation of the zero level set in the equation. The LS function is initialized as a signed distance function, which is zero at the free surface, negative in the air region and positive in the water region. In order to achieve smooth transition of the fluid properties from one fluid to the other, the free-surface is assumed to be of finite thickness defined by the LS function ϕ such that ϕ ≤ε , where ε is set to two or three characteristic grid lengths. The smoothed Heaviside function H (ϕ) can be defined as follows:

⎧ ⎪ H (ϕ ) = ⎨ ⎪ ⎩

0 1 [1 2

+

ϕ ε

+

πϕ 1 sin ( ε ) ] π

1

if ϕ < − ε ⎫ ⎪ if ϕ ≤ ε ⎬ ⎪ if ϕ > ε ⎭

Fig. 1. Schematic of determining the shortest distance from a node to the interface segment.Where xf is the foot of the perpendicular to a node xn, xs1 and xs2 are endpoints of interface segment, and xsm is the midpoint on an interface segment.

distance can be either the orthogonal distance (Fig. 1, case I) or the minimum distance between the node and two vertices of the interface segment (Fig. 1, case II). In order to reduce computational cost, the algorithm is conducted only for the nodes inside a narrow band where neighbor nodes around a cut element are defined. For more detail, see algorithm in Appendix A.

(6)

The density and viscosity are computed through the Heaviside functions as follows:

ρ (ϕ)=ρg +(ρl −ρg ) H (ϕ) μ (ϕ)=μg +(μl −μg ) H (ϕ)

(7) 2.3. Mesh adaptation

where the subscripts g and l denote gas and liquid, respectively. Since the advection equation of the LSM is of a hyperbolic type, a FEM with a suitable stabilization method needs to be employed to solve it. In this work, we adopted a least-square weighted residual method (LSWRM) (Choi, 2012) which has good mathematical properties such as natural numerical diffusion, and symmetry of the resulting algebraic systems for convective transport problems. The least-square weighted residual formulation of Eq. (5) can be written as follows (Choi, 2012): Given a velocity field u (x , t ) at t =t n+1, find ϕ n +1 (x , t )= ∑j ϕjn +1 wj∈H1 such that

⎞

⎞⎛

⎛

We have developed a local adaptive algorithm for efficient simulation of free-surface flows. This approach is based on our previous work (Sufyan et al., 2014). In this technique, the mesh is refined inside a narrow band around the free surface. The narrow band contains both cut elements and their neighbors (Fig. 2). We used level-set function as a criterion to mark the elements for mesh adaptation. The mathematical expression of the indicator used in this work is shown in Eq. (10). In the adaptation procedure, we firstly generate mid-nodes at the edges of each marked element. Then, each marked element is subdivided into four elements as shown in Fig. 3 (top row). The values of the variables at a newly created mid-node are copied from the previous adaptive mesh if that node coincides with a node existed in the previous adaptive mesh, or are obtained by linear interpolation. The second step is the treatment of hanging nodes in the neighboring elements. There can be one, two, or three hanging nodes in a neighboring element. The treatment of elements with one and two hanging nodes is shown in Fig. 3 (bottom row) while elements having three hanging nodes are treated as marked elements. After this step, the adapted mesh does not contain any hanging node and the smooth transition from refined to coarse region within the mesh is achieved. With the refinement of a marked element or neighbor element, the connectivity list is updated simultaneously. The new connectivity list is updated by changing the connectivity list of the father element and adding connectivity lists of children elements at the end of the connectivity list. It should be noted that mesh adaptation in our approach always starts from an identical initial mesh. Therefore, no extra step of coarsening is required. For more details, see (Ngo and Choi, 2016). The mesh after adaptation can be seen in Fig. 2b and Fig. 4. Lastly, it needs to be mentioned that efficient implementation of dynamic CSR (Compressed Sparse Row) method may be required for the present numerical method. However, the CPU time used for the construction of CSR format has been found to be quite small compared to the other parts: assemble of global matrix, solving the assembled matrix. Therefore, we have constructed a new CSR format whenever a newly adapted mesh is generated. It is also to be noted that the

∫Ω ⎜⎝w+∆t 12 un+1∙∇w⎟⎠ ⎜⎝ϕn+1+∆t 12 un+1∙∇ϕn+1⎟⎠ dΩ =

⎞⎛

⎛

⎞

∫Ω ⎜⎝w+∆t 12 un+1∙∇w⎟⎠ ⎜⎝ϕn−∆t 12 un∙∇ϕn⎟⎠ dΩ

(8)

w= ∑i αi wi∈H1.

In this study, the conjugate gradient (CG) method for all with incomplete LU preconditioning was used to solve Eq. (8). More details and comparisons of LSWRM with popular numerical approaches for advection equation can be found in the paper (Choi, 2012). 2.2.2. Direct re-initialization of level set method for an unstructured triangular mesh This work uses a direct technique to re-initialize the level set function for an unstructured triangular mesh (Ngo and Choi, 2015). This technique provides improved results in terms of mass conservation and is also very efficient in terms of computational overhead. In this approach, the interface is constructed from a collection of interface segments that are determined by the coordinates and level-set values of the vertices of the edges containing intersection points. The level set values at nodes near the interface are then replaced by the shortest distance from the node to the interface segments without changing its sign. The shortest distance is calculated using a geometrical formula that depends on the position of the foot of the orthogonal projection from a node onto the interface segment. Particularly, the shortest 205

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Fig. 2. Narrow band construction and adapted mesh inside the narrow band.

translation along the x-axis of the tank, is simulated in the present work. Two cases of the pure sway motion are considered: sway-base and sway-short. The time periods (T) are 1.94 s for sway-base, and 1.74 s for sway-short. In both cases, the filling level is 20% of the height and the amplitude is 0.06 m. A low filling level is selected for validation because sloshing with a low filling level produces more violent liquid motion inside the tank. Moreover, low filling conditions result in large impact pressure (Zou et al., 2015).

adaptation frequency is about 5 in the simulation of sloshing problems. 2.3.1. Adaptive mesh refinement for three-dimensional (3D) simulations For the extension of the adaptive algorithm to 3D case, the method proposed by (Ngo and Choi, 2016) has been employed. Each marked element will be subdivided into eight smaller elements (case I) while neighboring elements can be subdivided into two (case III), four (case II) or eight (case I) smaller ones as shown in Fig. 5. It should be noted that an iterative procedure is needed in their method for the treatment of hanging nodes until a final conformal mesh is achieved. For more details, see (Ngo and Choi, 2016). Fig. 6 shows the present 3D grid at four different instants with mesh adaptation near the free surface.

3.2. Selection of parameters 3.2.1. Grid dependency test Simulations are carried out on four triangular meshes G1–G4, whose details are presented in Table 1. In the table, n represents the number of nodes in the vertical direction of the base grid, while m represents the number of nodes in the horizontal direction of the base grid. The mean number of nodes with adaptive mesh refinement is calculated by dividing the sum of the total node count in each time step by the total number of time steps in the simulation. Mesh refinement is applied to the first three base grids G1–G3. In order to quantify the grid dependency of the solution, a time step of 0.001 s, which was proven to be sufficiently small as shown in Section 3.2.2, is selected for the simulations in this section. The impact pressure at P1 is selected to quantify the effect of grid resolution. Eq. (9) provides the mathematical definition of impact pressure. The grid dependence in terms of the relative difference ‘λ’ between G1A (G1 with adaptation) and G2A is found to be 6.85%. The relative

3. Results and discussion 3.1. Problem description Sloshing in a two-dimensional rectangular tank is considered as a benchmark problem. For the sake of validation, numerical results are compared with the experimental data of (Tanaka et al., 2000). Initial conditions in the present work are similar to those of the experimental setup, as shown in Fig. 7. The dimensions of the tank are 1.2 m wide (w) and 0.6 m tall. Pressure histories are recorded at positions, P1, P2 and P3 in the tank and compared with the experimental results. Visual comparison of the free surface between the numerical and experimental results is also presented. Pure sway motion, characterized by

Fig. 3. Schematic representation of division of marked element (1st row), treatment of neighbor elements with one hanging node (left, 2nd row) and two hanging node (right, 2nd row). “N” represents the total number of nodes during the division of current element.

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Fig. 4. Free-surface shapes with grid G2A at t=T/4, T/2, 3T/4, and T. The free-surface (ϕ=0) is represented by the thick solid line.

Fig. 5. Element subdivision process for 3D case.

smaller than the mean value. However, as time passes the flow inside the tank becomes violent and the total number of nodes exceeds the mean value.

difference between G2A and G3A was nearly 1%. Note that λ is the relative difference in the solutions of the two grids and is defined as

λ=

IPcoarser − IPfiner IPfiner

n

,

IP =

∑ (∫ i =1

0

Ti

Pdt )/ n (9)

3.2.2. Time-step dependency test Three different time step sizes are tested for the G2A grid. The corresponding results are compared in terms of pressure histories at P1. It can be seen from Fig. 9 that dt=0.0005 and dt=0.001 produce similar pressure histories, while for dt=0.0015 the peak pressure is lower. Therefore, throughout this study dt =0.001 is used for all numerical simulations. Fig. 10 compares the pressure histories between G3A and G4A grids at P1 and P3 obtained with the time step of 0.001 s. It can be observed that the pressure histories from aforementioned grids are comparable.

where IP represents the impact pressure and n represents the number of periods. Seven periods of sloshing were considered in the averaging calculations. The G4 grid has the same resolution as the baseline grid used in the study of (Rhee, 2005), who employed a structured grid of 60×120=7200 nodes to compare the numerical results with experimental data. In the present study, the G2A grid has approximately 5.47 (7200/1315=5.47) times fewer nodes than the Rhee's baseline grid. In the next section, we present a comparison of our present results with the results of (Rhee, 2005). Fig. 8 depicts the time history of the total number of nodes in the G2A mesh. The dashed line represents the mean number of nodes of the dynamically adapted mesh. In the first two periods, the liquid motion is not very violent and therefore the total number of nodes is

3.2.3. Size of adaptive band Two sizes of the adaptive band are used in this work. Two different criteria are used to designate the different adaptive bands. The 207

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Fig. 6. Free-surface shapes with 3D adaptation at (a) t=T/4, (b) t=T/2, (c) t=3T/4, and (d) t=T. The iso-surface (ϕ=0) represents the interface.

0.6m

air P3(1.2,0.15)

interface water

w=1.2m

P2(1.2,0.03) P1(1.17,0.00)

Fig. 7. Schematic of a 2D sloshing tank (all dimensions are in meters). Table 1 Details of computational grids. Grids

G1 G2 G3 G4

Nodes (n×m)

Number of nodes (Basegrids)

Mean number of nodes with Adaptation

Element size of base grid/w

08×15 15×30 30×60 60×120

164 688 2267 8552

341 1315 3556 –

0.075 0.04 0.02 0.01

Fig. 8. Time history of number of nodes of G2A grid.

following expressions describe the two criteria for the elements to be refined:

(i)

ϕ ≤ hi or

(ii ) ϕ ≤ 2hi

(10)

Where hi is the characteristic length of the element of the base grid near the free-surface. Fig. 11 shows a schematic of the adaptive band construction. It should be noted that the first indicator includes one layer of elements on each side of the cut elements (which contain the free surface). The first indicator refines a total of three element layers,

Fig. 9. Effect of time step size on pressure history of P1.

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Fig. 10. Comparison of pressure histories for G3A and G4A meshes at (a) P1 and (b) P3.

compared with experimental visualizations. The shapes are presented at the approximately same times within a period. The traveling wave can be observed in the numerical simulations, and the free surface profile qualitatively matches the experimental one. Fig. 14 shows the comparison of pressure histories at P1–P3 for the case of sway-base sloshing. The left column of the figure shows the comparison of G1A, G2A and G3A for the aforementioned points. The peak pressures from the G1A grid differ noticeably from G2A and G3A. The pressure histories from the G2A and G3A grids are similar. Hence, we selected the G2A grid for comparison with the experimental data. The right column of Fig. 14 shows the comparison of the present results with the experimental data. The numerical results are in good agreement with the experimental data, especially at P1. The overall profiles of pressure histories from the numerical simulations are similar to the experimental ones. However, the peak pressures at P2 and P3 are slightly overestimated. This is possibly because the mass of water increases in the present simulations as shown in Fig. 17. It should be noted that a disadvantage of the level-set approach is its lack of mass conservation. There are two pressure peaks in each period, which represent the impact of sloshing liquid when the traveling wave first hits the side wall, and then the effect of the liquid that had risen along the wall falling back down. The magnitude of first peak pressure is highest at P3 because the traveling wave, as discussed in the start of this section, hits the wall near P3. Furthermore, the pressure history at P3 shows zero pressure for approximately 0.7 T. This zero pressure indicates the instant when the numerical pressure tap is above the freesurface. The dashed lines (in the right column) represent the results obtained from the G3 base) grid. The results from the G3 and the G2A grids are similar. Therefore, it can be said that an adaptive solution is comparable to the non-adaptive solution with the same mesh resolu-

Fig. 11. Schematic representation of adaptive band size.

as can be seen in the schematic. The second increases the size of the adaptive band from three layers to five layers. Fig. 12 presents the effects of adaptive band size on pressure histories at P1 and P3. The 2hi adaptive band, which refines five grid layers including the cut element layer, produces similar results to those results of the 1hi adaptive band with respect to the magnitude of two peak pressures. Since the 1hi adaptive band is computationally more economical, all the simulations are based on the 1hi adaptive band size. 3.3. Comparison with experimental results In this section, we validate our method by comparing its solution of sway motions with the experimental results of (Tanaka et al., 2000). 3.3.1. Sway-base case In Fig. 13, the free-surface shapes from numerical simulations are

Fig. 12. Effect of adaptive zone size on pressure histories at (a) P1 and (b) P3.

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Fig. 13. Comparison of free surface shape among experimental and numerical results at two instants. (a)Experimental results (Tanaka et al., 2000), (b) G2A based simulation, and (c) G3 based simulation.

Fig. 14. Comparison of pressure histories at P1 (top), P2 (middle), and P3 (bottom) for sway-base case.

Bureau Veritas (Rhee, 2005). It is a simple and quite useful measure for off-shore structural design. The present results are slightly better than the results of (Rhee, 2005) in terms of average impact pressure. It should be noted that the present grid (G2A) is much coarser than the baseline grid of the aforementioned study. Fig. 15 demonstrates that the present results are similar to those (Rhee, 2005).

tion employed near the interface. The comparison of average impact pressure between the present computation and the literature is presented in Table 2. The average impact pressure is calculated by averaging the highest 10% of the pressure values, for the periods considered, which is over at least five periods. This averaging method is similar to the measure used by

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Table 3. The simulations with adaptation are about two times faster than the corresponding non-adaptive solutions. It should be noted that the G2 grid with adaptation and the non-adaptive G3 grid have the same grid resolution near the free surface. The computational efficiency is different for each base grid because it depends on many factors such as ratio of initial to mean-adaptive node count and the construction of dynamic storage by compressed sparse row (CSR).

Table 2 Comparison of the average impact pressure among present, experimental (Tanaka et al., 2000) and numerical (Rhee, 2005) results for sway-base case. Location

Experimental (kPa) (Tanaka et al., 2000)

Numerical (Rhee, 2005)

Present (kPa)

Error (%)

P1 P2 P3

1.533 1.355 1.398

1.490 1.471 1.345

1.521 1.466 1.366

0.78 8.19 2.28

3.4.2. Relative mass error Table 4 reveals that relative mass error strongly depends on the mesh resolution near the free surface. The relative mass error decreases with the increase in grid resolution near the free surface. Relative mass error history is presented in Fig. 17 for the cases of the sway-base (G2A and G3) and sway-short (G2A). For the case of sway-short the relative mass error is a little higher than for sway-base. The evolutions of impact pressure obtained from the present simulations are in a good agreement with those of the experiment (Tanaka et al., 2000). However, the peak pressures at P3 are slightly overestimated probably because mass error affects the solution to some extent. Therefore, an adaptive simulation with a finer base-grid needs to be conducted to obtain more accurate results with a smaller mass error for a violent sloshing simulation like in sway-short case.

3.3.2. Sway-Short The period of sloshing in the sway-short case (1.74 s) is shorter than the period in sway-base case (1.94 s). Fig. 16 depicts the comparison of pressure histories between experimental (Tanaka et al., 2000) and numerical results for P1-3. Computations for the sway-short case are carried out using the G2A grid. For this case, pressure histories for P1–P3 are also in good agreement with experimental data. 3.4. Comparison of adaptive and non-adaptive grids In this section, adaptive solutions are compared with non-adaptive solutions for the sway-base case described in Section 3.2 with respect to CPU time and relative mass error.

3.5. Effect of smoothing zone In the level-set method, the free surface is assumed to be of finite thickness (smoothing zone) to prevent sudden change of the fluid properties. A small size of the smoothing zone is desired to capture the sharp free surface. The size of the smoothing zone is defined by level-

3.4.1. Computational time The CPU times for adaptive and non-adaptive grids are presented in

Fig. 15. Comparison of present results with results of Rhee (2005).

Fig. 16. Comparison of pressure histories at P1 (left), P2 (middle), and P3 (right) for sway-short case.

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3.6. Effect of filling ratio on relative mass error

Table 3 Comparison of CPU time (second) among grids used in this study for sway-base case of 20% filling ratio. Numbers in parenthesis denote the grid resolution near the interface. Grid

Adaptive

Non-adaptive

G1 G2 G3 G4

81 (0.04) 351 (0.02) 1113(0.01) 3819 (0.005)

34 (0.075) 161 (0.04) 715 (0.02) 3056 (0.01)

In this section, the filling ratios of 40%, 60%, and 80% of height are studied with the amplitude of 0.06 m and the period of 1.94 s Fig. 20 shows the evolutions of the relative mass error at four different filling ratios. The relative mass error strongly depends on mesh resolution near the interface, as can be seen in Fig. 20(a and b). The relative mass error has been found to be less than 2% for all the filling ratios when G4A grid has been used. Fig. 21 shows the evolution of the absolute mass error (total loss/gain) at the four filling ratios for the simulations with G2A mesh. The absolute magnitudes of mass gain (loss) are similar except the case of the filling ratio of 60%, whereas relative mass error of 20% filling ratio is bigger than that of higher filling ratios for the simulations with G2A mesh. The relative mass error is the ratio of total mass loss (gain) to original mass. Therefore, the relative mass error of higher filling ratio becomes smaller since the original mass of higher filling ratio is bigger. Fig. 22 shows the pressure histories at P1 for filling ratios of 20%, 40%, 60% and 80% for G2A and G4A meshes. The peak pressure at P1 is highest for the filling ratio of 20% because the sway-base described in Section 3.2 is close to resonance. Sloshing of different filling ratios with the same sway-amplitude and period reveals that the peak pressure decreases with an increasing filling ratio. (Li et al., 2014) also found that peak pressure decreases with an increase in filling level, for the case of rolling motion of a tank. Fig. 23 shows pressure fields with hydrostatic pressure removed and free-surface motions inside the tank for 40%, 60% and 80% filling ratios. These images are taken at t=6T and t=6.25T. Adaptive mesh refinement can be observed in the figure for all three filling levels. The liquid motion is found to be less violent as the filling level increases. The results show that the present numerical approach can handle almost all the filling ratios of practical importance.

Table 4 Comparison of relative mass error (%) among grids used in this study for sway-base case of 20% filling ratio. Grid-resolution

Adaptive

Non-adaptive

0.04 0.02 0.01 0.005

31.25% 9.0% 7.1% 1.4%

30.31% 9.97% 6.82% –

set function ϕ such that |ϕ| ≤ε, where ε is set to 2 h with h being a element characteristic length near the free surface. In this section, numerical experiments are conducted with three different sizes of smoothing zone, 1.5hf, 2hf and 3hf, where hf is the element size (adaptive) near the free surface. The relative mass error is found to be 9.04%, 29.1%, and 36.3% for smoothing zone sizes of 1.5hf, 2hf, and 3hf respectively. It is found that the relative mass error of the sloshing simulation, where the evolution of free surface is violent, is sensitive to the size of the smoothing zone. Hence, a smoothing zone size of 1.5hf is selected for all simulations. Fig. 18 presents the effect of the smoothing zone size on the pressure history at P3. The profile of pressure history changes notably with an increase in smoothing zone size. Fig. 19 is presented to explain the problematic effect of the smoothing zone size on pressure history. From Fig. 19, it can be clearly seen that the pressure sensor at P3 point is positioned completely above the free surface in the case of 1.5hf, while the smoothing zone touches the P3 point in the case of the wider smoothing zone of 3hf. In the smoothing zone the material properties smoothly changes from air to water. Therefore, the wider smoothing zone signals the liquid presence to the sensor at P3 a little before and after the free surface crosses the P3 position. Consequently, the width of the pressure signal obtained from 3hf is bigger than that of 1.5hf as shown in Fig. 18.

3.7. Three-dimensional simulation of Sloshing In this section, we describe the results of adaptive mesh refinement based 3D sloshing simulations. The dimensions of 3D tank are 1.2 m wide (w), 0.6 m tall and 0.2 m deep, as used in the study of (Rhee, 2005). The 3D unstructured grid is a simple extension of G2 grid in the third direction with 30, 15, and 5 nodes along the edges of x, y, and z directions, respectively. The sway-base motion with 0.06 m amplitude and time period T=1.94 s is employed in the present 3D computations. According to the previous study by (Rhee, 2005), the free-surface shape looks nearly two dimensional except small areas near the side walls. The present result also confirms that free-surface shape is nearly twodimensional as shown in Fig. 24. The simulation of 20% filling ratio is performed with 3D adaptive mesh. Fig. 25(a) compares the pressure

Fig. 17. Comparison of relative mass error history for the cases of sway-short (G2A) and sway-base (G2A and G3).

Fig. 18. Effect of smoothing zone size on pressure history at P3 in the sway-base case.

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Fig. 19. schematic presentation of smoothing zone size for G2A grid (a) 1.5hf (b) 3hf.

Fig. 20. Relative mass error histories for four filling ratios. (a) G2A grid (b) G4A grid.

histories between the experimental data (Tanaka et al., 2000) and the present 3D numerical results. It shows that the second peak pressure is under-estimated because of the coarse resolution of base-grid. The relative mass error is less than 4% during the whole simulation of 8 periods as shown in Fig. 25(b). Further, another simulation is conducted for the case of 80% filling ratio. Free-surface shapes are almost two-dimensional as shown in Fig. 26. Fig. 27(a) compares the pressure histories between 2D and 3D adaptive solutions. It shows that the pressure histories of 2D and 3D simulations are almost identical. The relative mass error is also less than 3% for both 2D and 3D simulations as illustrated in Fig. 27(b).

4. Conclusion Numerical simulations of sloshing in a rectangular tank were carried out by implementing a dynamic mesh coarsening/refinement technique for an unstructured grid. A fractional four-step method was used to solve the incompressible Navier-Stokes equations. A level-set method was employed to capture the evolution of the free surface and

Fig. 21. Absolute mass error histories for four filling ratios.

Fig. 22. Pressure histories at P1 for four filling ratios. (a) G2A grid (b) G4A grid.

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Fig. 23. Free-surface shapes with pressure fields with hydrostatic pressure removed at two instants t=6T (left column) and t=6.25T (right column) for three filling ratios 40% (top), 60% (middle), and 80% (bottom).

surface. Two periods of sloshing were simulated and numerical results were found in good agreement with existing experimental data. It was found that the relative mass error strongly depended on the mesh resolution near the free surface. Application of the present adaptive approach reduced the computational cost by at least half. Although the relative mass errors of the present adaptive simulations were not negligible when the resolution of base-grid was not enough, the predicted impact pressures were in a good agreement with experimental data. The solutions of the adaptive method were similar to those of the non-adaptive method for identical interface resolution. It was also observed that the smaller smoothing zone size and the smaller adaptive band size produced accurate and efficient results and that the size of the smoothing zone notably affected the pressure histories, especially at the sensor above the initial free surface level. Fig. 24. 3D free-surface shape at a certain instant.

Acknowledgement the LSWRM was used to discretize the advection equation of the levelset method. A direct redistance method was used to re-initialize levelset as a signed distance function. With the present adaptation approach, the level set method accurately captured the sharp free

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF2014R1A2A2A01004879) which is gratefully acknowledged.

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Fig. 25. (a) Comparison of pressure histories between experimental data (Tanaka et al., 2000) and 3D numerical results. (b) Relative mass error history for 3D simulation.

Fig. 26. Free-surface shape at (a) t=6.4T and (b) t=6.8T from 3D simulation of 80% filling ratio.

Fig. 27. (a) Comparison pressure histories at P1 from 2D and 3D solutions (b) Comparison of relative mass error from 2D and 3D solutions for the case of 80% filling ratio.

Appendix A. Direct re-initialization algorithm for 2D triangular meshes Algorithm

Initialize: Set ϕ = δ , where δ is a big values for each cut elements, do Define the intersection point of the interface segment if (ϕo1⋅ϕo2 ≤ 0 ) then

xs = x1 −

ϕo1 (x ϕo1 − ϕo2 1

− x2)

end if Compute the shortest distance from a node of the narrow band to the interface segment 215

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for each node xn of the narrow band of the cut element, do if ( xf − xsm ≤ xsm − xs1 ), then

dx =

xn − xsm 2 − xf − xsm 2 (case I)

else dx = min{ xn − xsi } (case II) end if Update the LS value: ϕn = min{dx , ϕn} end for end for Recover the sign of LS function:ϕ = ϕ⋅sgn(ϕo ), where ‘sgn’ is a sign function, defined by: ⎧− 1ifϕo < 0 ⎪ sgn(ϕo ) = ⎨ 0ifϕo = 0 ⎪1ifϕ > 0 ⎩ o

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