Applications of An Eddy-Viscosity Eliminator Based on Sigmoid Functions in Reynolds-Averaged Navier-Stokes Simulations of Sloshing Flow

Reynolds-averaged Navier-Stokes (RANS) turbulence modeling can lead to the excessive turbulence level around the interface in two-phase flow, which causes the unphysical motion of the interface in sloshing simulation. In order to avoid the unphysical motion of the interface, a novel eddy-viscosity eliminator based on sigmoid functions is designed to reduce the excessive turbulence level, and the eddy-viscosity eliminator based on polynomials is extracted from the cavitation simulations. Surface elevations by combining the eddy-viscosity eliminators and classical two-equation closure models are compared with the experiments, the ones by using the adaptive asymptotic model (AAM) and the ones by using the modified two-equation closure models. The root-mean-squared error (RMSE) is introduced to quantify the accuracies of surface elevations and the forces. The relation between the turbulence level in the transition layer and RMSEs of surface elevations is studied. Besides, the parametric analysis of the eddy-viscosity eliminators is carried out. The studies suggest that (1) the excessive turbulence level in the transition layer around the interface has a significant influence on the accuracies of surface elevations and the forces; (2) the eddy-viscosity eliminators can effectively reduce the excessive turbulence level in the transition layer to avoid the unphysical motion of the interface; (3) the k − ω SST model combined with the eddy-viscosity eliminators is appropriate for predicting surface elevations and forces in RANS simulations of sloshing flow.


Introduction
Sloshing is the motion of the fluid with the interface between two phases inside a partially-filled tank. Many engineering problems involve with the sloshing phenomenon, such as the liquid cargo in the ships (Kim, 2001;Kim et al., 2004;Liu and Lin, 2008;Xue and Lin, 2011), the fuel tanks of the aircraft (Tang and Yue, 2017;Tang et al., 2019), the storage tanks in the earthquake (Goudarzi and Sabbagh-Yazdi, 2012) and so on. Since the awareness of the environmental problems rises, the demand for the clean energy, natural gas for instance, increases. Floating Liquefied Natural Gas vessel (FLNG) is the novel floating structure which can explore the natural gas resources in the ocean (Zhao et al., 2018). The liquefied natural gas (LNG) is stored in the internal tanks in FLNG at low temperature. There are various filling levels of the tanks which are excited by the ocean waves. Thus, the resonant sloshing is inevitable. Sloshing not only influences the ship motion, but also threatens the safety of the vessel structure.
The studies on sloshing can be divided into model tests, analytical solutions and numerical simulations. The analytical solution is based on the potential flow theory, but it is only suitable for the low-amplitude liquid sloshing in the tank with a simple geometry. The numerical simulations can be applied to the tank with arbitrary geometry. The numerical methods can be divided into the ones based on the potential theory and those based on the computational fluid dynamics (CFD). As the computing technique develops, CFD has been the popular method to simulate the sloshing flow. Maillard and Brosset (2009) and Karimi et al. (2015Karimi et al. ( , 2016 suggested that the gas phase plays a significant role in sloshing, and CFD can consider both the gas and liquid phases to solve the two-phase flow. Moreover, CFD can consider the viscosity and the turbulence effects. Lu et al. (2015) reported that the dissipative effects due to the viscosity have significant influence on the sloshing responses in both non-baffled and baffled tanks. Lu et al. (2015) also found that the sloshing amplitudes by using potential flow theory are over-predicted because the viscosity is ignored. Liu and Lin (2008) studied surface elevations of sloshing flow with different viscosities. Faltinsen (1978) and Zhao et al. (2018) added the artificial viscosity into potential flow theory to study sloshing flow.
Reynolds-averaged Navier−Stokes (RANS) simulations are the popular method to simulate sloshing flow. The computational cost in RANS simulation is much lower, compared with the large eddy simulation (LES) and direct numerical simulation (DNS). Furthermore, the requirements for the meshes in LES and DNS are much stricter than those in RANS simulations. Reynolds averaging means that a physical variable is decomposed into the time-averaged part and the fluctuation part. The Reynolds averaging on the governing equations leads to Reynolds stress term which cannot be computed directly. The RANS equations are closed based on Boussinesq hypothesis, and the eddy viscosity is introduced to represent the turbulence level. The RANS turbulence closure models are used to determine the eddy viscosity. Two-equation RANS closure models are widely used. However, classical two-equation RANS closure models can lead to the unphysical motion of the interface between the gas and liquid phase, such as the model, the model and the SST model. Classical RANS closure models were designed for single-phase flow. When the two-phase flow is considered as a single continuum fluid (the mixture), those classical RANS closure models can be applied to the two-phase flow. Mayer and Madsen (2000) reported the wave damping by using RANS models when the spilling wave breaking is simulated. Brown et al. (2014) evaluated different two-equation RANS closure models and concluded that the SST model is the one that performs the best for surface elevation predictions when the spilling wave breaking is studied. Brown et al. (2016) found that the RNG model is less accurate compared with the model which is less accurate compared with the SST model when the wave spilling and wave plunging were studied. Devolder et al. (2017Devolder et al. ( , 2018 reported that the surface elevation is significantly decreased over the length of the numerical wave flume and concluded that the excessive turbulence level around the interface brings the decrease of the surface elevation when classical RANS turbulence closure models are used. Larsen and Fuhrman (2018) evaluated several classical two-equation RANS closure models, such as the model, the RNG model, the model and the SST model, and proved that all those closure models are unconditionally unstable for the wave simulations regardless of the number of the phases being considered. In addition, in the simulation of the cavitation dynamics, Zwart et al. (2004), Frikha et al. (2008), Niedźwiedzka et al. (2016) and Long et al. (2017) found that the location of the interface between the gas and liquid phase around the cavitation cannot be determined accurately when RANS turbulence closure models are adopted.
Several studies have provided different solutions to improve the turbulence modeling. The interface between two phases are located by two advanced interface-capturing methods, volume-of-fluid (VOF) method and level set (LS) method (Bilger et al., 2017). Kamath et al. (2015) used an eddy-viscosity limiter to avoid the unphysical over-production of turbulence levels in the problem of wave-cylinder interaction by using LS method and the model. Grotle et al. (2017) adopted another eddy-viscosity limiter to study sloshing flow by using LS method and the model. Kamath et al. (2019) utilized the same eddy-viscosity as that in Grotle et al. (2017) to simulate free surface flows with LS method and the model. Devolder et al. (2017) proposed the buoyancy-modified SST model to prevent the interface from the unphysical motion when wave run-up around a monopile subjected to regular waves is simulated with VOF method. Larsen and Fuhrman (2018) proposed a new and formally stable closure model to simulate the surface waves with VOF method. Moreover, Zwart et al. (2004) and Long et al. (2017) adopted two modified formulas based on polynomials to compute the dynamic eddy viscosity, respectively, when cavitation dynamics was studied with VOF method. However, the researches about the effects of excessive turbulence level around the interface are still insufficient, particularly in the area of sloshing simulations. Moreover, the studies about the application of the turbulence modeling, especially for RANS turbulence modeling are insufficient in sloshing simulation. It is known that Kishev et al. (2006) and Gómez-Goñi et al. (2013) used the laminar model in their sloshing simulations. Lee et al. (2011), Liu andLin (2008) and Xue and Lin (2011) studied the sloshing flow using LES.
In this paper, in order to avoid the unphysical motion of the interface by using the Reynolds-averaged Navier− Stokes models, a novel eddy-viscosity eliminator (EVE) based on the sigmoid functions is designed to remove the eddy viscosity distributed in the transition layer where the liquid volume fraction varies from 0 to 1 but allows the eddy viscosity in the sub-regions where or 1 to pass. The eddy-viscosity eliminator based on the polynomials embedded in the modified formulas for computing the dynamic eddy viscosity in the cavitation simulation is extracted and introduced. In the sloshing simulations under the forced longitudinal and diagonal excitations, surface elevations by combining the eddy-viscosity eliminators with two-equation RANS closure models are compared with the experiments and the ones by using the adaptive asymptotic model (AAM), buoyancy-modified SST model and stable model. The effects of the excessive turbulence level around the interface on surface elevations and the forces are analyzed. The parametric analysis of eddy-viscosity elimin-ators are carried out.

Numerical model
The numerical model is implemented in the Open-FOAM, which is an open source CFD package. The finite volume method is used to solve the partial differential equations.

Governing equations ρ μ
In the two-phase flow solver of OpenFOAM (Open source Field Operation And Manipulation), the tank motion is treated with the dynamic mesh technique. The meshes move with the tank subjected to the excitations. The two phases are considered as a single continnum fluid (mixture) with the density and the dynamic viscosity . The finite volume method is used to discretize the integral form of the governing equations which are solved with respect to the earth-fixed reference frame. The integral form of the governing equations can be referenced to Li et al. (2019).

Volume of fluid (VOF) method ρ μ ν
The Volume-of-Fluid (VOF) method is used to capture the interface because VOF can ensure the mass conservation (Li et al., 2019). In the VOF method, an additional volume fraction field is used to capture the position of the interface. The density of the mixture, the laminar dynamic viscosity and the laminar kinematic viscosity are computed as: ρ = α 1 ρ 1 + (1 − α 1 ) ρ 2 ; (1) where and are the densities of the liquid and gas phases, respectively; and are the dynamic viscosities of the liquid and gas phases, respectively; and are the volume fractions for the liquid and gas phases, respectively.

Eddy-viscosity eliminators
When the two-phase flow is treated as a single continuum fluid, the transition layer where varies from 0 (the gas phase) to 1 (the liquid phase) is a density-stratification layer around the interface according to Eq. (1). The unphysical motion of the interface is attributed to the over-production of turbulence levels around the interface by using classical two-equation RANS closure models. The excessive turbulence level in the transition layer has the negative influence on the motion of the interface. Spedding et al. (1996) and Sarkar (2003) reported that the turbulence level in the density-stratification layer is suppressed in physics, for example, the buoyancy effect can restrict the velocity fluctuations. In order to match the fact in physics and suppress the turbulence level in the transition layer in the simulation, an eliminator is devised to eliminate the overestimated eddy viscosity in the transition layer, but allow the α 1 = 0 α 1 = 1 φ eddy viscosity in the sub-regions where or to pass. The eddy-viscosity eliminator is inserted into the following relation where is the eddy-viscosity eliminator; the kinematic eddy viscosity is determined by RANS turbulence closure models; is the dynamic eddy viscosity, and it is used in the momentum equation. The eddy-viscosity eliminator can be used with the two-equation turbulence closure models implemented for two-phase incompressible flow in Open-FOAM. Zwart et al. (2014) adopted a modified formula to compute the eddy viscosity when the cavitation flow is simulated by using the model. The modified formula is written as: Long et al. (2017) adopted another formula to compute the eddy viscosity when the cavitation shedding dynamics is studied with the RNG model.

Eddy-viscosity eliminator based on the polynomials
The eddy-viscosity eliminators can be extracted from Eq. (9) and Eq. (10) based on Eq. (8), and they contain the polynomials. The eddy-viscosity eliminators can be rewritten as: When Eq. (5) is used, it can be proved that Eq. (11) and Eq. (12) are equivalent. Fig. 1 shows the eddy-viscosity eliminator based on Eq. (12) corresponding to different values of , such as 1, 2, 5, 10, 15 and 20. As the order of the polynomials increases, more amount of eddy viscosity is eliminated in the transition layer considering the term , which suggests that more excessive turbulence level around the interface is eliminated.

Eddy-viscosity eliminator based on the sigmoid functions
The novel eddy-viscosity eliminator is proposed in this section based on the sigmoid function where the scaling parameter controls the transition width; the translation parameter controls the transition position along the -axis. Fig. 2a shows the sigmoid function with different scaling parameters. As the scaling parameter a decreases, the transition width of the sigmoid function decreases. φ The eddy-viscosity eliminator is designed as: where the scaling parameter ; the two translation parameters , . Fig. 2b illustrates the eddyviscosity eliminator .

Modified two-equation closure models
In order to reduce the overestimated eddy viscosity in the simulations of surface waves, Devolder et al. (2017) proposed the buoyancy-modified SST model, and Larsen and Fuhrman (2018) proposed the stable model. Both studies adopt VOF method to capture the interface. The buoyancy-modified SST model has two modifications, compared with the SST model implemented for two-phase incompressible flow in OpenFOAM (Devolder et al., 2017): (1) the mixture density is explicitly implemented in transport equation; (2) a buoyancy term is added into the right-hand side of the transport equation of turbulent kinetic energy. The stable model has three main modifications, compared with the model implemented of two-phase incompressible flow in OpenFOAM (Larsen and Fuhrman, 2018;Grotle et al., 2017): (1) the mixture density is explicitly implemented in transport equations; (2) a buoyancy term is added into the right-hand side of the transport equation of turbulent kinetic energy; (3) the eddy vis-λ 1 λ 2 cosity is computed by using the limiters which involve and in Larsen and Fuhrman (2018). ρ The mixture density is implemented in the transport equations to consider the variation of the mixture density across the interface. The buoyancy terms in the buoyancymodified SST model and the stable model are proportional to , but they have different coefficients. N is Brunt-Väіsälä frequency or buoyancy frequency. The implementation of and the addition of the buoyancy term together reduce the overestimated turbulence level in the transition layer and improve the accuracies of surface elevations. The closure model which excludes the buoyancy term from the transport equation of turbulent kinetic energy of the buoyancy-modified SST model is used to study the influence of the mixture density on the accuracies of surface elevations, and that closure model is denoted as the SST model with in this paper.
Moreover, the model, model and SST model denote the closure models implemented for twophase incompressible flow in OpenFOAM in this paper, respectively. The mixture density is not implemented in the transport equation of the model (Launder and Spalding, 1983), model (Grotle et al., 2017;Devolder et al., 2018;Kamath et al., 2019) and SST model (Devolder et al., 2017).

Setup of the numerical computations
The model tank is a cube with the length m. The origin of the earth-fixed Cartesian coordinate system is at the center of the tank bottom before the tank moves, as shown in Fig. 3. The wave probes w3 and w5 are illustrated. The model tests were carried out by Faltinsen et al. (2005), the ullage gas is air, and the liquid is water. The initial wa- The natural circular frequency is where and are the natural mode numbers along -axis and -axis, respectively, . The wave number Fig. 1. Variation of eddy-viscosity eliminator based on the polynomials with respect to in the transition region. Fig. 2. (a) Scaling of the sigmoid function ; (b) Variation of eddy-viscosity eliminator based on the sigmoid functions with respect to .
The tank moves under the forced longitudinal excitation and forced diagonal excitation. The forced longitudinal excitation is a a/L = 0.00871 ω e ω e /ω 1,0 = 1.037 ω 1,0 = 6.86 where is the amplitude with ; is the excited circular frequency in rad/s, , rad/s, rad/s. The forced diagonal excitation is where and are the amplitudes along -axis and -axis, respectively, and , , rad/s, rad/s.
The eddy-viscosity eliminators are implemented based on ESI-OpenFOAM. The numerical simulations are carried out by using the interDyMFoam solver. The implicit Euler scheme is adopted to discretize the transient term. The High Resolution (HR) scheme is used to handle the convection term. The Gauss linear corrected scheme is adopted to treat the Laplacian term. The PIMPLE algorithm in OpenFOAM is used to decouple the governing equations. The boundary conditions at the tank walls for the liquid volume fraction , the flow velocity and the pressure in excess of the hydrostatic pressure are zeroGradient, movingWallVelocity and fixedFluxPressure, respectively. The wall functions are used in RANS turbulence modeling at the boundary walls. The air density and water density are 1.2 kg/m 3 , 998 kg/m 3 , respectively. The kinematic viscosity of air and the kinematic viscosity of the water are m 2 /s and m 2 /s, respectively. The surface elevation is computed at runtime according to the distribution of liquid volume fraction , which can be referenced by Devolder et al. (2017). The meshes around the interface have been refined, and the smallest grid spacing is 0.005 m. The time step is ajustable according to the maximum Courant number which is set to be 0.2, and the initial time step is 0.0001 s. The grid convergence has been verificated.

R
The root-mean-squared error (RMSE) is used to quantify the accuracy of the signal computed by using the numerical model comparing with the experiment.
where is the -th element of the numerical signal; is the -th element of the experimental signal; is the total number of the elements.

Surface elevations under the forced longitudinal excitation
The two eddy-viscosity eliminators and are first combined with the SST model. The results are compared with the experiment, the ones by using the laminar model and the AAM method, as shown in Figs. 4 and 5. At the initial time interval before 6 s, all time histories almost coincide with each other because it takes some time for the turbulence level to develop. The time history of the surface elevation by using the SST model cannot agree with the experiment after 6 s. From 6 s to 13 s in Fig. 4, the crests and the troughs by using the SST model are attenuated compared with the experiment, and the phase lags behind. It is the results of the excessive damping based on RANS turbulence modeling. The excessive damping is caused by the excessive turbulence level in the transition layer around the interface. As the excitation continues, the surface elevation by using the SST model increases again from 13 s. The excessive turbulence level around the interface based on RANS turbulence modeling and the excitation bring the surface elevation to another orbit to develop, which causes the unphysical motion of the interface.
The results by using the SST model combined with and agree with that of the experiment. The differences among the results by using the SST model with two eddy-viscosity eliminators are not easy to distinguish from Figs. 4 and 5. It suggested that the surface elevation can be better predicted when the excessive turbulence level around the interface is reduced by the eddy-viscosity eliminators ) and . The phase of the surface elevation computed by using the laminar model can agree with the experiment, however, the crests and the troughs by using the laminar model are larger than those of the experiment. The turbulence modeling which is represented by the eddy viscosity in RANS simulations considers the turbulence cascade. Since is inserted into the momentum equation, the diffusion of the momentum is enhanced. When there is no turbulence modeling in the laminar model, the diffusion of the momentum is decreased, therefore the interface is less restricted.
The time history by using the SST model combined with is compared with those by using the buoyancy-modified SST model, the stable model and the SST model with in Fig. 6 and Fig. 7. As shown, those surface elevations agree better with the experiment than that by using the SST model shown in Fig. 4 and Fig. 5. However, the comparison of the accuracies of surface elevations is not easy just from Fig. 6 and Fig. 7. The RMSEs of surface elevations will be shown in the following.
The SST model is a blend of the model and the model to address the weaknesses of the model and the model (Brown et al., 2014). Surface elevations computed by using the model without/with the eddy-viscosity eliminator and the model without/with the eddy-viscosity eliminator are compared with that of the experiment in Fig. 8. The surface elevation by using the model with can agree with the experimental results till 8 s, whereas the surface elevation by using the model agrees with the experimental results before 6 s. Both surface elevations are attenuated seriously. The surface elevations by using the model without and with can agree with the experimental results before 6 s, and the surface elevation by using the model with is attenuated later than that by using the model. The eddy-viscosity eliminator cannot improve the accuracies much, and it is attributed to the defections of the model and The RMSEs of surface elevations at wave probe w3 under the forced longitudinal excitation are listed in Table 1. It is found that (1) the RMSEs of surface elevations by using the model, the model and the SST model which are combined with the eddy-viscosity eliminators are reduced. The surface elevations can be better predicted when the excessive turbulence level around the interface is reduced; (2) the RMSE of the surface elevation by using the laminar model is smaller than those by using the model, the model and the SST model, which suggests that the appropriate RANS turbulence modeling is important; (3) the eddy-viscosity eliminator performs better than , and the RMSEs of the surface elevation by using the SST model with is the smallest among the models in Table 1; (4) the RMSE of surface elevation by using the stable model is less than that by using the buoyancy-modified SST model. The RMSE of surface elevation by using the buoyancy-modified SST model is less than that by using the SST model with . The RMSE of surface elevation by using the SST model with is much less than that by using the SST model. The combination of explicit implementation of in transport equations and the addition of the buoyancy term can reduce the RMSE of surface elevations effectively; (5) the eddy-viscosity eliminator combined with the SST model and two modified two-equation closure models are both effective to reduce the RMSEs of surface elevations.
As shown from 25 s to 32 s in Fig. 5 and Fig. 7, the time histories of surface elevations by using the above models cannot agree with the experimental results, including AAM method which is a single-phase method based on potential theory. There may be some unknown mechanisms about the energy transfer between the frequency components, and those mechanisms are not considered by the present models.

Surface elevations under the forced diagonal excitation
Surface elevations by using the SST model combined with the eddy-viscosity eliminators and are compared with the experimental results, the ones by using the SST model and the AAM method, as shown in Fig. 9 and Fig. 10. The data from the experiment and the results by using the AAM method are available from 35 s to 58 s provided by Faltinsen et al. (2005). Surface elevations before 7 s are almost the same because the turbulence level needs some time to develop. At each descending stage of the envelope of the surface elevation, the crests and the troughs by using the SST model are attenuated ahead of time, and then surface elevations by using the SST model increase again after they decrease to some level as the excitation continues. The surface elevation seems to fall into another orbit to develop by the combination effect of the excessive turbulence level around the interface and the excitation. The surface elevations by using the SST model with and agree with the experimental data. The crests and the troughs by using the laminar model and the AAM method are larger than the experimental data at the descending stage of the surface elevation.
As shown in Fig. 11, surface elevations by using the SST model with , the buoyancy-modified SST model, the stable model and the SST model with are compared with the experimental data. Those surface elevations agree better with the experimental data than that by using the SST model illustrated in Fig. 9 and Fig. 10.
Surface elevations by using the model without/with the eddy-viscosity eliminator and the model without/with the eddy-viscosity eliminator are compared with the experimental data in Fig. 12. As shown, all surface elevations cannot agree with the experimental data.

Effects of turbulence level around the interface ν
The averaged value of the eddy viscosity distributed in the transition layer from to is computed, and the averaged is a measure of the turbulence level in the transition layer around the interface. As for the sloshing flow under the forced longitudinal excitation, the averaged values of by using different models are compared in Figs. 13a−13d, and the -axes of those figures are in the logarithm scale. As shown in Fig. 13a, the averaged k − ω value of by using the model is larger than that by using the model, and the averaged value of by using the model is larger than that by using the SST model. Those averaged values of are larger than the laminar kinematic viscosity of the liquid phase and the laminar kinematic viscosity of the gas phase by several orders of magnitude. The averaged values of increase a lot from 1 s to 5 s because the fluid in the tank is excited from the rest, and it takes some time for the turbulence level to develop. From Figs. 13b−13d, it can be found that the eddyviscosity eliminators effectively reduce the averaged values of in the transition layer. When sloshing flow is under the forced longitudinal excitation, by comparing Fig. 13 and Table 1, it is found that there is a corresponding relation between the excessive turbulence levels in the transition layer around the interface and the RMSEs of surface elevations. The lower averaged values of tend to correspond to smaller RMSEs of surface elevations. In Figs. 13b and 13c, the lower averaged values of by combining the eddy-viscosity eliminators correspond to the smaller RMSEs of surface elevations by using the model and the model. In Fig. 13d, the averaged value of by using the SST model with is smaller than that by using the SST model with , and the RMSE of surface elevation by using the SST model with is smaller than that by using the SST model with , as shown in Table 1. The results support the conclusion by Brown et al. (2014) that the SST model perfoms the best for the wave predictions.
As for the sloshing flow under the forced diagonal excitation, the findings are consistent with those under the forced longitudinal excitation, which supports the conclusion that there is the corresponding relation between the excessive turbulence levels in the transition layer around the interface and the RMSEs of surface elevations.

Influences on the hydrodynamic loads
Sloshing is an interfacial flow, and the interface has a significant influence on the hydrodynamic loads. The hydrodynamic loads are computed by the integration of the pressure and the viscous stress on the entire walls of the tank. The component of the hydrodynamic load along theaxis is denoted as the longitudinal force , and the component of the hydrodynamic load along the -axis is denoted as the transverse force . The forces are important in the studies, such as the coupling analysis between the internal tank sloshing and the ship motion.
For the simulations under the forced longitudinal and diagonal excitations, the longitudinal force by using the SST model with and are compared with the experimental data and the ones by using the laminar model, the SST model and the AAM method, as shown in Figs. 14−16. As expec- Fig. 11. Time histories of surface elevations at wave probe w5 from 35 s to 58 s under the forced diagonal excitation by using different methods, including the SST model with , the SST model with , the buoyancy-modified SST model and the stable model.

470
LI Jin-long et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 463-474 k − ω ted, the forces by using the SST model with the eddyviscosity eliminators agree better with the experimental data than those by using other models.

Parametric analysis of the eddy-viscosity eliminators
In order to study the influence of the parameters and in the eddy-viscosity eliminator and the parameter in , different values of the parameters are chosen and listed in Table 2. The corresponding eddy-viscosity eliminators are plotted in Fig. 17. The corresponding eddy-viscosity eliminators are shown in Fig. 1. The simulations are carried out under the forced longitudinal excitation. The different eddy-viscosity eliminators are combined with the SST model.
Four cases are devised to allow different amount of turbulence levels in the transition layer to pass. Cases A, B and C correspond to the eddy-viscosity eliminator , and Case D corresponds to the eddy-viscosity eliminator . Case A fixed the parameter to be 0.01, and is changed to allow more amount of eddy viscosity in the transition layer near the liquid side to pass from case A1 to A5. Case B fixed the parameter to be 0.99, and is changed to allow more amount of eddy viscosity in the transition layer near the gas side to pass from case B1 to B5. In Case C, the parameters and are changed simultaneously and symmetrically to allow more amount of eddy viscosity in the transition layer to pass from Cases C1−C5. In Case D, the parameter is changed to allow more amount of eddy viscosity in the transition layer near the liquid side to pass from Cases D1 to D5.
The RMSEs of surface elevations of different cases in Table 2 are compared in Fig. 18. It can be found that as less amount of eddy viscosity in the transition layer are allowed to pass, surface elevations agree better with the experimental data, and the RMSE of surface elevation decreases. The RMSEs of surface elevations corresponding to Case B are smaller than those corresponding to Case A, which suggests that the overestimated eddy viscosity in the transition layer near the liquid side may contribute more to the un- physical motion of the interface compared with the eddy viscosity in the transition layer near the gas side. The RMSEs of surface elevations corresponding to Case A and Case B are smaller than those corrersponding to Case C because more amount of eddy viscosity in the transition layer for Case C are allowed to pass comparing with Case A and Case B. Furthermore, three horizontal lines are plotted in Fig. 18 to represent three different RMSE levels, including the RMSEs of surface elevations by using the SST model, the laminar model and the SST model with eddy-viscosity eliminator . The RMSE of surface elevation by using the SST model with eddy-viscosity eliminator is the smallest of all, which is consistent with the finding in Figs. 4 and 5. The RMSE of surface elevation by using the laminar model is larger than that by using the SST model with , which is attributed to the absence of turbulence modeling. The RMSE of surface elevation by using the SST model is larger than that by using the laminar model, which is caused by the excessive turbulence level in the transition layer around the interface in RANS turbulence modeling. The eddy-viscosity eliminator corresponds to Case D3. As shown in Fig. 18, as increases in Case D, the RMSEs of surface elevations decrease to approach a limit which is a little larger than the RMSE of surface elevation by using the SST model with . When the eddyviscosity eliminator is combined with the SST model, the amount of the eddy viscosity near the gas side and the liquid side are eliminated asymmetrically. The eddy viscosity in the transition layer near the gas side is eliminated much more than that near the liquid side, and the elimination of the eddy viscosity in the transition layer near the liquid side depends on the value of .

Conclusions
RANS turbulence modeling can bring the excessive turbulence level. The excessive turbulence level around the interface and the excitation together lead to the unphysical motion of the interface in RANS simulations of sloshing flow. In order to solve the problem and match the fact in   Table 6.
physics that the turbulence level in the density-stratification flow is suppressed, a novel eddy-viscosity eliminator based on the sigmoid functions is devised to reduce the turbulence level in the transition layer, and the eddy-viscosity eliminator based on polynomials embedded in the modified formula for computing the dynamic viscosity in the cavitation simulations is extracted and introduced. Both eddy-viscosity eliminators can be combined with the classical two-equation closure models implemented for two-phase incompressible flow in OpenFOAM, such as the SST model, and the transport equations of those closure models are not modified. In order to compare different solutions, two modified two-equation closure models are introduced into the sloshing simulations, including the buoyancy-modified SST model and stable model in which the transport equations are modified, e.g., the explicit implementation of mixture density and the addition of the buoyancy term. φ(α 1 ; q 1 = 0.01, q 2 = 0.99) φ(α 1 ; n = 10) k − ω φ(α 1 ; q 1 = 0.01, The simulations are carried out under the forced longitudinal and diagonal excitations, and the results of surface elevations and forces are compared with the experimental data. The eddy-viscosity eliminators can effectively reduce the excessive turbulence level in the transition layer by using VOF method. Thus, the RMSEs of surface elevations are reduced, and the accuracies of surface-elevation predictions are improved. The novel eddy-viscosity eliminator performs better than . The SST model combined with 0.99) performs the best in the predictions of surface elevations compared with other models in this paper. The buoyancy-modified SST model and stable model are also the effective solutions to avoid the unphysical motion of the interface in the computations. For the model and model, the adoption of the eddy-viscosity eliminators can reduce RMSEs of surface elevations, but the surface elevations still cannot agree with the experimental data. The reason may be attributed to the defections of the and models themselves. φ(α 1 ; n) n φ(α 1 ; q 1 = 0.01, q 2 = 0.99) k − ω φ(α 1 ; q 1 = 0.01, q 2 = 0.99) The corresponding relation between the RMSEs of surface elevations and the turbulence level in the transition layer is found, and the smaller turbulence level in the transition layer tends to correspond to the smaller RMSE of the surface elevation. Moreover, the surface elevations have a significant influence on the hydrodynamic loads, and the accuracies of computed forces can be much improved by using the eddy-viscosity eliminators. In the parametric analysis of the eddy-viscosity eliminators, the RMSEs of surface elevations by using approach a limit when increases, and the RMSE of the surface elevation by using is smaller than the limit. The SST model with the eddy-viscosity is appropriate to compute surface elevations and the forces in RANS simulations of sloshing flow by using VOF method.