Hybrid SW-NS SPH models using open boundary conditions for simulation of free-surface flows

Ocean Engineering 196 (2020) 106845

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Hybrid SW-NS SPH models using open boundary conditions for simulation of free-surface flows Xingye Ni a, b, *, Weibing Feng a, Shichang Huang b, Xin Zhao b, Xinwen Li b a b

College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing, China Zhejiang Institute of Hydraulics & Estuary, Hangzhou, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Smoothed particle hydrodynamics Hybrid model Open boundary condition Shallow water equation Navier-Stokes equation

Balancing the trade-off between the simulation detail and the computational efficiency in a hydrodynamic nu­ merical model is generally difficult, e.g., one can hardly find a single set of governing equations that is capable of efficiently describing the wave propagation processes from offshore to nearshore and accurately presenting the details of wave breaking and interacting with coastal structures simultaneously. To address this predicament, open boundary coupling schemes are proposed in this paper to hybridize a SWE (shallow water equations)-based SPH (Smoothed Particle Hydrodynamics) model with a NS (Navier-Stokes) equations-based SPH model. The NSSPH model is used to present the nonlinear hydrodynamic features in the main study domains, while the remaining domains are simulated with the SWE-SPH model, which calculates much faster than the NS-SPH model. The hybrid SPH models are applied to simulate solitary wave propagation, steady flows over a bump, dam breaking waves and standing waves. The hybridization performance on the information transmitting, mass and energy conservation, simulation detail, computational accuracy, computational efficiency and computa­ tional convergence speed is fully validated and evaluated. The results indicate that the proposed open boundary coupling schemes are capable of improving the computational efficiency on the premise of computational ac­ curacy and simulation detail.

1. Introduction The SPH (Smoothed Particle Hydrodynamics) method (Gingold and Monaghan, 1977; Liu and Liu, 2003; Monaghan, 1994) is a promising meshfree Lagrangian numerical method that shows advantages over conventional Euler mesh-based methods on the aspects of moving boundaries, free surfaces and multiphase interface capture and particle tracking. During the most recent decade, the SPH method has been widely applied to simulate highly nonlinear hydrodynamic problems (Gotoh and Khayyer, 2018; Violeau and Rogers, 2016; Zhang et al., 2017) with complex water surfaces and fluid-structure interactions, e.g., hydraulic jumps (Federico et al., 2012), tidal bores (Ni et al., 2018a,b), nearshore wave shoaling (Li et al., 2012), wave runup and breaking on beaches (Dalrymple and Rogers, 2006; Lo and Shao, 2002) and waves interacting against coastal facilities (Altomare et al., 2017; Ni and Feng, 2014; Liu et al., 2014). Great efforts have been made on the theoretical study of SPH method, including numerical stability (Antuono et al., 2010; Marrone et al., 2011), conservation and error analysis (Gotoh et al., 2014; Gui

et al., 2015; Khayyer et al., 2017a; Khayyer et al., 2017b; Wang et al., 2019; Sun et al., 2019; Violeau and Fonty, 2019), solid boundary con­ ditions (Ferrand et al., 2013; Fourtakas et al., 2019), open boundary conditions (Ni et al., 2018a,b; Wang et al., 2019), etc. Nevertheless, one of the drawbacks of the SPH method is its low computational efficiency, and the main solutions at present include high performance computing (HPC) (Domínguez et al., 2013; Rustico et al., 2014), particle refinement techniques (Barcarolo et al., 2014; Vacondio et al., 2013) and hybridi­ zation techniques, which are discussed in this paper. Hybrid SPH models can be divided into two categories according to the purpose of hybrid­ ization. One category of hybrid SPH models uses various numerical models to describe the corresponding movements of various subjects in the same system, e.g., using an SPH model to simulate the nonlinear water movements inside a tank while using an FEM (Finite Element Method) model to simulate the interaction between an elastic plate and the sloshing water (Yang et al., 2012; Khayyer et al., 2018). Another example (Altomare et al., 2014a,b) is using an SPH model to simulate nearshore water waves while using a DEM (Discrete Element Method) model to simulate the armor blocks on sea breakwaters and studying the

* Corresponding author. College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing, China. E-mail address: [email protected] (X. Ni). https://doi.org/10.1016/j.oceaneng.2019.106845 Received 8 September 2018; Received in revised form 26 November 2019; Accepted 8 December 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Sketch of hybrid SPH models in the literature: (a) layer coupling model, (b) wave paddle coupling, (c) overlapping zone coupling.

Fig. 2. Sketch of two-way open boundary coupling scheme.

wave runup features. However, the aforementioned hybrid SPH models will not be discussed here. The second category of hybrid SPH models, which are the study topic in this paper, aim to improve the computa­ tional efficiency on the premise of computational accuracy and simu­ lation detail by using SPH models to present the nonlinear hydrodynamic features in the main study domains while using con­ ventional mesh-based methods or simplified models to simulate the rest of the system. This category of hybrid SPH models can be further divided into two types: a layer coupling model (LCM) and a section coupling model (SCM). Navier-Stokes (NS) equations are generally adopted as the gov­ erning equations by the submodels of LCMs, which are arranged up and down, as shown in Fig. 1-a. The upper submodel uses the SPH method to

discretize the governing equations to capture the nonlinear water sur­ faces, while the lower submodel uses conventional Euler meshes to improve computational efficiency. The two-way coupling scheme, also known as the strong coupling scheme, is applied to connect the two submodels. Since the evolution of the free surface is not involved with the coupling boundaries, the two-way coupling schemes for LCM are relatively easy to handle. For example, Marrone et al. (2016) proposed a hybrid model that connected a WCSPH (Weakly Compressible SPH) model and an FVM (Finite Volume Method) model, while the hybridi­ zation of an ISPH (Incompressible SPH) model and an FVM model was implemented by Napoli et al. (2016). Section coupling models are more common in the applications of hybrid numerical wave flumes, which use NS equations-based SPH 2

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effect, leading to eventual coupling failure. Additionally, the two aforementioned coupling schemes do not suit the flow conditions because the schemes limit particle creation and particle deletion operations. By contrast, open boundary coupling bears no such problems. The positions of the open boundaries remain steady during the simulation, similar to conventional mesh-based models, introducing a Euler perspective into the Lagrangian system. Few SCMs have been reported using open boundary coupling schemes, probably due to the difficulties in implementing open boundaries with varying free surfaces in an SPH model, considering its Lagrangian features. Kassiotis et al. (2011) implemented a one-way hybridization of an FDM-based Boussinesq wave model and an SPH model and simulated the wave runup on a block-protected breakwater. However, the incident waves will be re­ flected to the offshore side after interacting with coastal structures, and an original open boundary is not capable of transmitting the reflected waves back to the mesh-based submodel on the offshore side. Instead, second-reflections will be induced at the open boundaries, resulting in wave energy accumulation in the nearshore submodel and incorrect information synchronization between submodels. Thus, the hybrid model becomes unstable, and the results of long-term simulation are unreliable. Hence, further work should be conducted on the modifica­ tions and improvements of the open boundary coupling scheme. In this paper, a two-way open boundary coupling scheme and a modified one-way open boundary coupling scheme using a nonreflective open boundary are proposed to hybridize a one-dimensional SWE-based SPH model with a vertical two-dimensional NS equations-based SPH model. The paper is structured as follows: The categories and research progress of hybrid SPH models are reviewed in Section One. The sub­ model implementations and the open boundary coupling schemes are introduced in Section Two. The two-way coupling performance of the proposed hybrid SPH model is validated using a solitary wave propa­ gation test case in Section Three. The hybrid model is then applied in Section Four to simulate steady flows over a bump, dam breaking waves runup on a slope, dam breaking waves impacting against a vertical wall and standing waves, further verifying the modeling accuracy and the computational efficiency. Sections Five and Six present the conclusions and acknowledgements, respectively.

Fig. 3. Sketch of submodel communication for two-way open bound­ ary coupling.

submodels to simulate the nearshore wave evolutions and the detailed wave-structure interactions (e.g., wave breaking and overtopping), while mesh-based numerical methods are used to solve nonlinear SWEs (shallow water equations) or Boussinesq equations and to describe the wave propagations from offshore to nearshore. According to the coupling schemes that connect the submodels, SCMs generally include wave paddle coupling, overlapping zone coupling and open boundary coupling. Wave paddle coupling (Fig. 1-b) is the most widely used scheme. Early in 2010, Narayanaswamy et al. (2010) hybridized the Boussinesq equation-based FUNWAVE model (Kirby et al., 1998) with the SPHysics model (Gomez-Gesteira et al., 2012) using a two-way solid wave paddle and validated the hybrid SPH model with a solitary wave propagation case. Later, Altomare et al. (2014a,b, 2015) implemented a one-way hybridization between the SWASH model (Zijlema et al., 2011) and the DualSPHysics model (Crespo et al., 2015). Deformable wave paddles were adopted to adjust to the velocity distributions in various water depths. Similarly, Verbrugghe et al. (2018) used deformable wave paddles to connect the OpenWave3D model (Engsig-Karup et al., 2009) and the DualSPHysics model, and it was a two-way hybridization. Different from wave paddle coupling, which transports water surface fluctuation with moving boundaries, overlapping zone coupling (Four­ takas et al., 2017; Sriram et al., 2014) transports the velocities and pressures through the submodels by overlapping zones, as shown in Fig. 1-c. The overlapping zones transform with the movement of the water body, and there are no particle creations or particle deletions among the submodels. As the mass transportation phenomenon will be induced by the nonlinear nearshore waves in the propagation directions, the afore­ mentioned coupling boundaries, including wave paddles and over­ lapping zones, will gradually deviate from the initial positions with the simulation progress. Moreover, deformable coupling boundaries, e.g., deformable wave paddles and deformable overlapping zones, will suffer from the growing larger transformations caused by the velocity differ­ ence between the upper water and the lower water. These trans­ formations will be strengthened by the nonlinear mass transportation

2. Hybrid models and coupling methodology In conventional section coupling models, the submodels on the nearshore side are generally implemented based on NS equations and solved with the SPH method, while mesh-based wave models are chosen to be the submodels on the offshore side. A similar nearshore submodel is adopted here, named NS-SPH, while the offshore submodel is also an SPH model that is based on one-dimensional linear SWEs, named SWESPH accordingly. This is the first hybrid SPH model that combines two SPH submodels using different governing equations. To implement the information and mass exchange between SWE-SPH and NS-SPH, two open boundary coupling schemes are proposed in this paper, including a two-way coupling scheme and a one-way coupling scheme based on a

Fig. 4. Sketch of one-way open boundary coupling scheme using FOBC. 3

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tensor with � ταβ ¼ υt ρ 2Sαβ

2 kδαβ 3

�

2 ρCI Δ2 δαβ jSαβ j2 3

(4)

where k is the SPS turbulent kinetic energy. The turbulent viscosity eddy

coefficient υt ¼ ðCs ΔÞ2 jSj , where the Smagorinsky constant Cs ¼ 0.12. δαβ ¼ {1 α ¼ β; 0 α6¼β} and Δ is the particle initial interval. jSj ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sαβ Sαβ and Sαβ is the element of the SPS strain tensor. � � 1 ∂uα ∂uβ Sαβ ¼ þ (5) 2 ∂xβ ∂xα Then, Eq. (3) can be written into SPH form as ! N N X X Dui Pj Pi 4υ0 mj rij ⋅uij ri Wij mj 2 þ 2 ri Wij þ g þ ¼ �� �2 Dt ρ ρ j i ρi þ ρj �rij � j¼1 j¼1 ! N X τj τi þ mj 2 þ 2 ri Wij

Fig. 5. Sketch of submodel communication for one-way open boundary coupling using FOBC.

nonreflective open boundary condition. The implementation of the two submodels and the coupling schemes are introduced successively.

The core concept of the SPH method is discretizing the study subject into a group of particles that not only have material properties, e.g., density, velocity and pressure, but also work as computational nodes. A field function fðrÞ (Eq. (1)) and its derivative r⋅fðrÞ (Eq. (2)) at any location r in the system can be calculated by accumulating the contri­ butions from all the neighboring particles with a weighting function, called the kernel function. Readers are suggested to refer to the works of Liu and Liu (2003) for a more detailed theoretical discussion and formulation. N X mj j¼1

ρj

� f rj Wij

ρi

(6)

The NS-SPH model follows the assumption of WCSPH, and the δ-SPH formula (Antuono et al., 2010; Marrone et al., 2011) is adopted here to calculate the change rate of fluid particle densities:

2.1. NS-SPH model

〈f ðri Þ〉 ¼

ρj

j¼1

N N X dρi X mj mj uij ⋅ri Wij þ δhc0 ψ ⋅ri Wij ¼ dt ρj ij j¼1 j¼1

〈rρ〉Li ¼

h

� r �rij �

ρi � ij�2

ψij ¼ 2 ρj

N X

ρj

〈rρ〉Li þ 〈rρ〉Lj

�

i

(7) (8)

(9)

ρi Li ri Wij dVj

j¼1

"

(1)

N X

rj

Li ¼

� ri � ri Wij dVj

#

1

(10)

j¼1

〈r⋅f ðri Þ〉 ¼

N X mj j¼1

ρj

� f rj ri Wij

(2)

where the second term on the right-hand side of Eq. (7) is the dissipative term for the density change rate. The dissipative term helps reduce the unphysical oscillation of fluid density and pressure in WCSPH, where δ controls the dissipative intensity. δ ¼ 0.1 in this paper. The fluid pressure relates to its density explicitly based on the assumption of artificial compression. Thus, the pressure can be calcu­ lated by the equation of state directly as �� �γ � ρ P¼B 1 (11)

where f is an arbitrary field function in the system. The subscripts i and j denote the central particle and the neighboring particles, respectively. m and ρ denote the mass and density of a particle. The kernel function Wij ¼ W(ri-rj, h), where h is the smoothing length, presents the weighting influence of particle j on particle i. The momentum equation of the Navier-Stokes equations can be written into the Lagrangian form as Du ¼ Dt

1 1 rP þ g þ υ0 r2 u þ r⋅τ

ρ

ρ

ρ0

where γ ¼ 7 and B ¼ c20ρ0/γ. ρ0 is the reference density, and we have ρ0 ¼ 1000 kg/m3 in the water simulation. c0 is the artificial speed of sound, which is generally larger than 10 times the reference velocity to main­ tain the change rate of fluid density within 1%, theoretically. The reference velocity in this paper is calculated using (gd)0.5, where g is the gravity acceleration and d is the water column height in the dam

(3)

where the third and the fourth terms on the right-hand side denote the laminar viscosity term and the SPS (Sub-Particle Scale) turbulent vis­ cosity term (Gotoh et al., 2004) based on the LES (Large Eddy Simula­ tion) model. ν0 is the kinetic viscosity, and τ is the SPS turbulent stress

Fig. 6. Sketch of the solitary wave propagation case. 4

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Fig. 7. Water depth evolutions in the solitary wave propagation case: (a) SWE-SPH, (b) NS-SPH, (c) NS-SPH→SWE-SPH, (d) SWE-SPH→NS-SPH

breaking case or the initial water depth in the numerical wave flume case.

Lagrangian form (Xia et al., 2013) as

2.2. SWE-SPH model The shallow water equations can be obtained by integrating the Navier-Stokes equations in the vertical direction. The linear SWEs without the Coriolis force term and viscosity term are written in the

Dd ¼ Dt

dr⋅u

(12)

Du ¼ Dt

gðrd þ rbÞ þ gSf

(13)

where d, b and u denote the water depth, bed elevation and the depth-

Fig. 8. Velocity field evolutions in the NS-SPH→SWE-SPH solitary wave propagation case. 5

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Fig. 9. Pressure field evolutions in the NS-SPH→SWE-SPH solitary wave propagation case.

Fig. 11. Density evolutions near the coupling open boundary.

Fig. 10. Enlarged snapshots of velocity field and pressure field near the coupling open boundary at t ¼ 2.85 s.

averaged horizontal velocity, respectively. Sf denotes the bed friction contribution and is calculated by Eq. (14), where n is the Manning coefficient. Sf ¼

n2 ujuj d4=3

(14)

Then, Eq. (13) is written in SPH form to obtain the acceleration of particle i: Dui ¼ Dt

N X

Vj ri Wij

g j¼1

g

N X Vj j¼1

dj

bj

� bi ri Wij

N X

Vj Π ij ri Wij j¼1

g

n2 ui jui j d4=3 i (15) Fig. 12. Convergence speed of the hybrid SPH model. 6

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Fig. 13. Steady flows over a bump simulated by various models: (a) SWE-SPH model, (b) Hybrid model (5.0 m
Fig. 14. Horizonal velocity distributions near the hydraulic jump.

where Vj denotes the volume of neighboring particle j. Vj/dj in the SWESPH model has a similar meaning as mj/ρj in the NS-SPH model, i.e., the system space that is occupied by particles j (area for 2D and length for 1D). The artificial viscosity term, which helps stabilize the calculation, is obtained by � 8 > < αμij cij 2μij uij ⋅rij < 0 dij Π ij ¼ (16) > : 0 uij ⋅rij � 0

According to the mass conservation and the incompressible assumption, the volume of each particle in SWE-SPH remains constant, indicating that the smoothing length reduces as the water depth in­ creases, or the smoothing length increases as the water depth reduces. Thus, the variable smoothing length is obtained by � �1=Dm d0 (17) hi ¼ h0 di where h0 and d0 are the initial smoothing length and the initial water depth of particle i, respectively. Dm is the system dimension. The modified formula for water depth integration (Eq. (18)) pro­ posed by Xia et al. (2013), which helps reduce error in the presence of steep topography, is adopted to calculate the water depth d and the smoothing length h together with Eq. (17).

where uij and rij are the relative velocity and the relative position be­ tween the central particle i and the neighboring particles j, respectively. uij ¼ ui-uj and rij ¼ ri-rj. cij and dij are the mean speed of sound and the

mean water depth, respectively, with cij ¼(ciþcj)/2 and dij ¼(diþdj)/2, pffiffiffiffiffiffi and ci ¼ gdi . μij ¼ hij uij∙rij/r2ij, where hij is the mean smoothing length

N X

with hij ¼(hiþhj)/2. The artificial viscosity coefficient α in this paper equals 1.0, which makes a satisfactory balance between the numerical stability and the numerical dissipation.

di ¼

Vj Wij i¼1

N X Vj i¼1

dj

bi

� bj Wij

(18)

The particle’s bed elevation bi is calculated using the approach 7

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Fig. 15. Vertical velocity distributions near the hydraulic jump.

where Nb is the bed particle number in the supporting domain and bbj , hbj and ωbj denote the bed elevation, the smoothing length, and the occupied space (area for 2D and length for 1D) of bed particle j, respectively. κi is the modification coefficient of the smoothing function, given as

Table 1 Computational efficiency comparisons for the case of steady flows over a bump. Case

Numerical model

NS-SPH domain (m)

Particle number

CPU time (s)

A

SWE-SPH

None

433 (1.2 � 10 2Np) 5991 (0.17Np) 15526 (0.44Np) 35284 (1.00Np)

15 (3.84 � 10 4T) 6120 (0.16T) 16292 (0.42T) 39233 (1.00T)

B C D

Hybrid-SPH Hybrid-SPH NS-SPH

[5.0, 7.0] [3.5, 8.5] [0.0, 10.0]

Nb X

κi ¼

Nb � 1 X ωbj bbj W ri κi i¼1

� rbj ; hbj

rbj ; hbj

(20)

i¼1

2.3. Coupling methodology In the most widely used wave paddle coupling scheme for SPH hy­ bridization, the fluid movement information should be first transformed into the movement of a solid paddle, which is then used to drive the fluid in the other submodel. In contrast, the open boundary (Ni et al., 2016, 2018a,b) coupling scheme (Fig. 2) proposed in this paper reduces the intermediate steps and makes the information exchange between sub­ models more direct and more expedite. The submodel communication for two-way open boundary coupling is sketched in Fig. 3, and the scheme works as follows: For the SWE-SPH model, the water depth dSWE

suggested by Vacondio et al. (2012a,b), in which a set of fixed bed particles are distributed in a Cartesian grid and the bed elevation of particle i is interpolated from the bed particles in its supporting domain, defined as bi ¼

�

�

ωbj W ri

(19)

Fig. 16. Sketch of dam breaking waves runup on a slope. 8

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are the limitations of wave paddle coupling and overlapping zone coupling. Conventional section coupling models are generally used in non­ breaking wave cases, and they theoretically do not suit breaking wave cases due to the limitation of coupling schemes. It is discovered that numerical instability will also be triggered when breaking waves prop­ agate through the aforementioned two-way open boundary and some­ times this results in the model blowing up. Thus, a modified one-way open boundary coupling scheme based on the nonreflective FOBC (Flather’s Open Boundary Condition) (Ni et al., 2016, 2018a,b) is pro­ posed here as a supplement for the two-way open boundary coupling scheme; see Fig. 4. The computational domain of SWE-SPH covers the entire domain of the hybrid model, considering the CPU time of SWE-SPH can be neglected when compared with that of NS-SPH. Similar to the two-way open boundary coupling scheme, the water depth dSWE and the depth-averaged velocity USWE at xSWE are also interpolated from the neighboring SWE fluid particles and transmitted to FOBC. The buffer zone particles are updated following the rules of FOBC and drive the internal fluid movement in NS-SPH, as shown in Fig. 5. When the re­ flected waves from beaches or coastal structures interact with the coupling boundaries, those waves will be transmitted outward of the system thoroughly owing to the nonreflective features of FOBC. Hence, the second-reflection that occurred in front of the original open boundaries by Kassiotis et al. (2011) will be efficiently prevented. Practical application cases (Section 4.2 in this paper) show that the modified one-way open boundary coupling scheme acts as a satisfactory connection between SWE-SPH and NS-SPH when breaking waves or breaking bores propagate through. It should also be noted that the time step in NS-SPH is used as the time step for SWE-SPH since the former is much smaller than the latter. Moreover, the CPU time rate between SWE-SPH and NS-SPH is approximately 10 4, so the computational efficiency of the hybrid model is mainly controlled by the corresponding parameters in NS-SPH, e.g., particle numbers and speed of sound. 3. Solitary wave propagation through two-way coupling boundaries

Fig. 17. Free water surface evolutions at four gauges: (a) x ¼ 1.4 m, (b) x ¼ 3.4 m, (c) x ¼ 4.0 m, (d) x ¼ 4.5 m.

Nonreflective energy transmitting and unrestricted mass trans­ portation between submodels are the main criteria for a qualified hy­ bridization scheme. In this section, a solitary wave propagation case is used to validate the proposed coupling scheme. The flume with a flat bed in Fig. 6 is 5.0 m in length and 0.4 m in depth. The solitary wave with H ¼ 0.1 m is generated by the open boundary on the left-hand side of the flume and propagates to the FOBC on the right end. The reason why the solitary wave propagation case is chosen is that unlike the oscillation movement patterns of regular waves, all water particles in a solitary wave travel in the same direction as the wave. This feature guarantees the monodirectional transport of fluctuations and mass among the submodels and makes it easy to identify if there is a reflection when the transport travels from a submodel to another. A SWE-SPH model (Case A) and a NS-SPH model (Case B) are used as the control groups to simulate the solitary wave case separately, as shown in Fig. 7-a and Fig. 7-b. Since the linear SWEs are applied as the governing equations in SWE-SPH, the nonlinear feature of solitary waves makes the wave profile gradually lean forward into a bore shape. In contrast, NS equations give a better description of solitary wave propagation, during which the wave profile remains symmetrical and stable. In Case C and Case D, the flume is separated into two parts at x ¼ 2.5 m, and different submodels are hybridized to validate the performance of the coupling schemes when solitary waves propagate from NS-SPH to SWE-SPH or from SWE-SPH to NS-SPH. Fig. 7-c shows that the solitary wave propagates through the coupling boundaries without reflections, although there is a refraction of the wave peak trajectory due to the use of different governing equations. In the case of SWE-SPH→NS-SPH

Table 2 Relative error comparisons for free water surface calculation. Case

Numerical model

Gauge #1 (x ¼ 1.4 m)

Gauge #2 (x ¼ 3.4 m)

A B C

SWE-SPH Hybrid-SPH NS-SPH

5.50% 5.50% 4.69%

5.58% 5.64% 5.26%

and the depth-averaged velocity USWE at xSWE are interpolated from the neighboring SWE fluid particles and transmitted to NS-OBC (open boundary condition). The buffer zone particles (dashed circles) in the NS-OBC are redistributed according to the new water depth, and their velocities are updated as well, driving the internal fluid movement in the NS-SPH. Meanwhile for the NS-SPH model, the free surface elevation at xNS is captured to obtain the water depth dNS, and the depth-averaged velocity UNS is integrated from the bed to the water surface. The two variables are transmitted to SWE-OBC and are used to update the ve­ locity and water depth properties of the SWE buffer zone particles (gray circles), completing the loop of information feedback from NS-SPH to SWE-SPH. The width of the open boundary is 4 times the particle initial inter­ val, and the influence on the hybridization can be neglected. Since the two submodels are both capable of creating and deleting particles through their own open boundaries independently, the proposed hybrid model can transmit not only surface waves but also water flows, which 9

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Fig. 18. Dam breaking wave evolutions on the slope: (a) wave runup, (b) breaking bores, (c) wave plunging on the slope, (d) undular bores.

If water particles periodically oscillate around the coupling bound­ aries, e.g., regular waves, nonneglectable numerical noise will probably be induced by the reflection phenomena. Better results are expected if the NS-SPH model is hybridized with nonlinear SWE-based or Boussi­ nesq equation-based numerical models. During the time 2.5 s
Table 3 Computational efficiency comparisons for the case of breaking waves runup on a slope. Case

Numerical model

Particle number

CPU time (s)

A B C

SWE-SPH Hybrid-SPH NS-SPH

452 (0.017Np) [5560, 22506] ([0.21, 0.87]Np) 25864 (Np)

31 (9.2 � 10 4T) 17150 (0.51T) 33715 (1.00T)

(Fig. 7-d), a slight reflection is observed at the coupling boundaries. This indicates that the two-way coupling of NS-SPH and SWE-SPH is actually not equal in one direction with another. In the open boundary of NS-SPH, there is no other hydrodynamic information except for water depth and depth-averaged velocity, e.g., the vertical distribution of horizontal velocity and pressure and the vertical velocity of water particles. In the open boundary of SWE-SPH, water depth and depth-averaged velocity are the only two variables that can be provided. When the information is transmitted from NS-SPH to SWE-SPH, although parts of the hydrodynamic details are lost during the coupling process, there is still enough information for the calculation of linear SWEs. However, when the information is transmitted in the opposite direction, there is not enough information for the complete calculation of NS-SPH. Some basic assumptions must be made on the coupling open boundaries, e.g., zero vertical velocities and hydrostatic pressure, leading to the aforementioned reflection. 10

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Fig. 19. System mass evolutions of the hybrid model.

Fig. 20. Sketch of dam breaking waves impacting against a vertical wall.

(ρ0 ¼ 1000 kg/m3) due to the weakly compressible feature of WCSPH. The deeper the water is, the larger the density becomes. When the sol­ itary wave propagates over the density sensors, the measured densities vary with the evolutions of water depth. Although there are some fluc­ tuations on the density evolution curves, the general performance is satisfactory as the change rate of fluid density is kept within 1% in line with the numerical assumption. The convergence speed of the hybrid SPH model is also evaluated using three test cases with different particles resolutions of 0.02 m, 0.0133 m and 0.01 m. The L2 norm of the water depth error is shown in Fig. 12 with a satisfactory order of convergence of 1.52 for NS-SPH submodel and 1.16 for SWE-SPH submodel.

4. Hybrid model applications 4.1. Steady flows over a bump Shallow Water Equations are commonly applied to simulate free surface water flows, which have a much larger horizontal scale than vertical scale. The SPH models (Vacondio et al., 2012a,b; Xia et al., 2013) based on SWEs have proven that they are capable of not only addressing generally shallow water flows but also efficiently capturing flow shocks and automatically identifying wetting-drying boundaries. Steady flows over a bump are widely used as one of the benchmark test cases (Ni et al., 2016; Vacondio et al., 2012a,b) for SWE models: the subcritical flows from the upstream are turned into a supercritical state after flushing over a bump, and a hydraulic jump is then formed between the bump and the subcritical flow in the downstream, as shown in Fig. 13-a. In this paper, a 1D slip channel with a length of 10 m and a 11

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Fig. 21. Pressure evolutions of two sensors on the vertical wall: (a) P1 (3.22 m, 0.16 m), (b) P2 (3.22 m, 0.584 m).

Fig. 22. Snapshots of pressure fields solved by hybrid-SPH model at (a) t ¼ 1.24 s and (b) t ¼ 1.82 s.

bump height of 0.2 m is used. To increase the simulation difficulty, the bump width is set to 0.2 m, which is only half of that value in general cases. The bed elevation of the channel is defined by Eq. (21) as � 0:2 0:2ðx 5Þ2 x 2 ½4; 6�m bðxÞ ¼ (21) 0 x 62 ½4; 6�m

two models will be discovered around the hydraulic jump behind the bump. The reason is that the basic assumptions of SWEs do not hold true for the local flows around steep topography, so that the SWE-based models fail to handle the violent hydraulic features of velocity distri­ bution and free surface variations. An accurate description of the highly nonlinear hydrodynamic phenomenon near the hydraulic jump, including water body collapsing, water particle splashing and backflow zone, can only be achieved using NS equations. On the other hand, if the whole computational domain is simulated by NS-SPH, most of the CPU power will be wasted on the upstream and downstream channels with slowly varying water levels, which actually has been very well described by SWE-SPH. Hence, in this section, the NS-SPH model is applied to simulate the transcritical flow and the hydraulic jump near the bump, while the SWE-SPH model is utilized to simulate the rest of the domain. Meanwhile, submodels are hybridized with a two-way open boundary

The inflow velocity in the upstream is 0.435 m/s, and the outflow depth in the downstream is 0.330 m. The numerical resolution of SWESPH and NS-SPH is 0.02 m and 0.01 m, respectively. The simulation time of the case here is 25 s, and Fig. 13 shows the flow velocities when the system comes into a steady state. The comparisons between the nu­ merical results and the analytical solutions of SWEs indicate that SWESPH is capable of predicting the transcritical flows with a hydraulic jump, see Fig. 13-a. However, if the test case is resimulated using the NSSPH model (Fig. 13-d), deviations of numerical results between these 12

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Fig. 23. Free water surface evolutions at four gauges: (a) x ¼ 2.725 m, (b) x ¼ 2.228 m, (c) x ¼ 1.73 m, (d) x ¼ 0.6 m.

Fig. 24. Sketch of a standing wave case.

coupling scheme to improve the computational efficiency on the premise of computational accuracy and simulation detail. Fig. 14 and Fig. 15 present the horizontal velocity distributions and the vertical velocity distributions at four sections (x ¼ 4.5 m, 5.5 m, 6.2 m and 7.5 m) near the hydraulic jump. The circles denote the results of NS-SPH (Fig. 13-d), while the solid lines denote the results of the hybrid model (Fig. 13-c). Figs. 14 and 15 show that the proposed hybrid model matches the full-scale NS-SPH model in the corresponding computa­ tional domain. Additionally, the computational efficiency of various models is compared in Table 1, where Np and T denote the particle number and the CPU time used in Case D with full-scale NS-SPH. It is indicated that 1) the SWE-SPH CPU time is negligible, 2) the hybrid model CPU time is remarkably reduced compared with the full-scale NSSPH, and 3) the CPU time is generally proportional to the used particle number. All the aforementioned results have proven that the proposed hybrid SPH model reaches the same level of computational accuracy and simulation detail in modeling highly nonlinear hydrodynamic problems compared with the full-scale NS-SPH model, while achieving consider­ able improvement in the computational efficiency.

4.2. Dam breaking waves runup on a slope Dam breaking and wave runup are two classic problems in hydro­ dynamic simulations, which both involve nonlinear water surface evo­ lutions and wetting-drying boundaries. In this section, the two cases are joined to form a test of dam breaking waves runup on a slope, which is applied to further verify the performance of the proposed open bound­ ary coupling scheme. At the initial state (Fig. 16-a), a rectangle water body with a height of 0.25 m and a length of 2.25 m is deployed on a flat bed. A solid wall is set on the left of the water body, and a 1:10 slope is set at x ¼ 3.40 m. The water body collapses due to gravity after the simulation starts. The breaking wave runs up along the slope and turns the kinetic energy into potential energy, as shown in Fig. 16-b. The water front on the slope then slips down and runs toward the offshore side in the form of breaking bores. Later, the water body oscillates be­ tween the solid wall and the slope until the mechanical energy is dissi­ pated, and the water ultimately calms down on the bed. Extreme nonlinear water movements (e.g., dam breaking waves and breaking bores) transmitting through coupling boundaries are involved in this test case, resulting in much larger numerical difficulty than 13

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Fig. 25. Free water surface evolutions at the nodes and antinodes of standing waves: (a) xd ¼ 0, (b) xd ¼ L/4, (c) xd ¼ L/2, (d) xd ¼ 3L/4.

results and the experimental data (De Leffe et al., 2010). The relative errors are calculated by Eq. (22), where h0 is the initial water depth of 0.25 m. Generally, the numerical results of the three models agree with the experimental data satisfactorily. NS-SPH yields a slightly smaller error than SWE-SPH, and the hybrid model yields an equivalent level of error with the other two models. There are no experimental data at Gauge #3 and Gauge #4, while the relative errors between Hybrid-SPH and NS-SPH at the two gauges are 3.70% and 2.65%, respectively, indicating good agreement in the prediction of water wave evolutions on the slope. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u � N � u1 X ηa ηb 2 (22) εðηa ; ηb Þ ¼ t N 1 h0 The general propagations of breaking waves, e.g., runup on the slope (Fig. 18-a), can be well predicted by the SWE-SPH model. However, due to the limitation of the simplified governing equations, the SWE-SPH model cannot provide detailed nonlinear water movements with frac­ tured surfaces, e.g., the breaking bores induced by the water slipping down from the slope (Fig. 18-b), the plunging waves on the slope (Fig. 18-c) and the undular bores propagating toward the shore (Fig. 18d). In contrast, the proposed hybrid model not only inherits the excellent capability of simulating complex hydrodynamics problems from the NSSPH model but also improves the computational efficiency remarkably. As shown in Table 3, the particle number in the hybrid model ranges from 5560 to 22506 during the 20 s simulation, and the mean number is approximately 15154, which is only 44.9% of the particle numbers required in NS-SPH, and requiring only 50.9% of the original CPU time. In addition to computational accuracy, computational efficiency and simulation detail, the conservation of system mass is another important performance index that should be considered for hybridization schemes. Especially in cases with water particles traveling around the coupling boundaries, unobstructed mass transportation and qualified conserva­ tion of system mass are the key factors in long-term simulation. The oneway open boundary coupling scheme is potentially more likely to lose its balance of system mass compared with a two-way scheme. Fig. 19

Fig. 26. The envelopes of the standing waves in the NS-SPH submodel.

nonbreaking regular wave propagation cases simulated by conventional hybrid SPH models. Even if the two-way open boundary coupling scheme is applied, the hybrid model is still not robust enough to bear the shock from the breaking waves due to the huge mismatch between the information provided by different submodels. Hence, the modified oneway open boundary coupling scheme is utilized in this section to address this challenging problem. The coupling boundary is set at x ¼ 2.00 m, the right side of which is modeled with NS-SPH, while SWE-SPH covers the entire computational domain. The computational resolution is 0.005 m, and the simulation time is 20 s. This test case was also simu­ lated by De Leffe et al. (2010) using another SWE-based SPH model, and the water surface evolutions at x ¼ 1.4 m and x ¼ 3.4 m were validated with experimental data. In this paper, two more gauges are deployed on the slope at x ¼ 4.0 m and x ¼ 4.5 m to compare the performance of various numerical models. Similar to the case of steady flows over a bump, three different nu­ merical models are used here, including SWE-SPH, Hybrid-SPH and NSSPH. The free water surface evolutions at 4 gauges are validated in Fig. 17, and Table 2 compares the relative errors between various model 14

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Fig. 27. Pressure field snapshots of the standing waves during one wave period: (a) t ¼ t0, (b) t ¼ t0þT/4, (c) t ¼ t0þT/2, (d) t ¼ t0þ3T/4.

presents the time evolutions of the system mass during the simulation, indicating that the mass of the two submodels fluctuates as the water body oscillates on the bed. The total mass of the hybrid model generally remains steady with a mean fluctuation rate of 3.0%. The mass fluctu­ ation gradually decreases as the simulation progresses, and the total mass converges to the eventual theoretical value. Considering that only the one-way coupling scheme is used in the case that suffers from multiple shocks from the breaking waves, the mass conservation per­ formance of this modified one-way coupling model is acceptable.

The laboratory experiment of this dam breaking case was conducted by Buchner (2002) and the impacting force on the vertical wall was measured by two pressure sensors which were located at P1 (3.22 m, 0.16 m) and P2 (3.22 m, 0.584 m), respectively. Four water level gauges were also deployed at x ¼ 2.725 m, 2.228 m, 1.73 m and 0.6 m to measure the free surface evolutions. The pressure evolutions of P1 and P2 by hybrid-SPH model and NSSPH model are presented in Fig. 21. The numerical results are shifted by 0.327 s to compare with experimental data (Buchner, 2002). The two-peak type pressure evolutions at P1 is well simulated by SPH models and the peak values are generally satisfactory. The pressure at P2 is mainly the result of hydrostatic pressure, which is also well predicted owing to the accurate calculation of water front uprising. The pressure results given by hybrid-SPH model and NS-SPH model generally match with each other, indicating that the coupling scheme has acceptable error on the force calculation. Snapshots of pressure fields solved by hybrid-SPH model at t ¼ 1.24 s and t ¼ 1.82 s are also provided in Fig. 22, where fluid particles in NS-SPH model are color-mapped and free surface particles in SWE-SPH model are presented with black solid circles. The free water surface evolutions at four gauges are validated in Fig. 23, which shows a good match between the numerical results and experimental data before the reflected waves arrive at the gauges around t ¼ 2.0 s. Since multiple water-air interfaces are formed due to the plunging waves and the top surface is detected as the free water surface, the numerical water levels of H1, H2 and H3 during t ¼ 1.7 s–2.5 s are unauthentic and they are generally larger that experimental data. After the water movement gradually calms down, the numerical results

4.3. Dam breaking waves impacting against a vertical wall Another classic case of dam breaking waves impacting against a vertical wall, which is often used in the simulation of tsunami or green water, is applied in this section to validate the force calculation of hybrid-SPH model. As shown in Fig. 20-a, a rectangle water body with a height of 0.60 m and a length of 1.20 m is deployed on the left part of a water tank with a length of 3.22 m at the initial state. The dam breaking waves run downstream along the flat bed and impact against the vertical wall on the right side of the tank. Then the waves are reflected and propagate backward with violet and breaking free surface. Similar to the dam breaking case in Section 4.2, the one-way open boundary coupling scheme is also used here with a coupling boundary at x ¼ 1.00 m. The entire computational area is modeled by SWE-SPH, while the NS-SPH only calculates right part of the tank. The test case is also modeled with NS-SPH as comparisons for the hybrid-SPH results. The particle resolution of both hybrid-SPH model and NS-SPH model is set as 0.005 m. 15

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and the experimental data agree with each other again.

model the water movements in the offshore areas and provide a better simulation of the nonlinear wave propagation process from the offshore to the nearshore.

4.4. Standing waves

Acknowledgements

A simple standing wave flume with a length of 2L ¼ 5.00 m and water depth of d ¼ 0.4 m is simulated as the last case in this paper to evaluate the energy conservation performance in long term run, using hybridSPH model with one-way open boundary coupling scheme. The computational domain of SWE-SPH covers the entire flume while NSSPH calculates the domain from x ¼ L to x ¼ 2L, as shown in Fig. 24. Regular waves with wave period T ¼ 1.45 s, wave height H ¼ 0.04 m and wave length L ¼ 2.50 m are generated on the left part of the flume using an open boundary wave generation algorithm (Ni et al., 2016). The incident waves propagate through the coupling open boundary and get reflected by the vertical wall at x ¼ 2L. Then standing waves are induced due to the interaction between the reflected waves and the incident waves. This case runs for 15 wave periods with particle resolution of 0.01 m. Fig. 25 shows the free water surface evolutions at xd ¼ 0, L/4, L/2 and 3L/4, respectively, where xd denotes the distance from the targeted section to the vertical wall. The water levels vary in a simple harmonic pattern with wave heights of 2H at the antinodes of standing waves (xd ¼ 0 and xd ¼ L/2) as predicted by analytical solutions, while water surface almost remains steady at the first node from the vertical wall (Fig. 25-b). The water surface at the second node from the vertical wall (Fig. 25-d) varies in a regular pattern probably due to the influence of skew waves which propagate from SWE-SPH and has been discussed in Section 3. Nevertheless, the skewness vanishes after waves propagate in NS-SPH as shown in Fig. 26, while Fig. 27 provides pressure field snapshots of the standing waves during one wave period. This standing wave case also indicates that energy conserves satisfactorily in the proposed hybrid-SPH numerical wave flume, which is not only robust in long term run, but also remains a steady mean water level.

This work is supported by the National Natural Science Foundation of China [No. 51809238 and No. 51779228], the Fundamental Research Funds for the Central Universities [No. 2019B06514] and the Scientific Research Program of Zhejiang Institute of Hydraulics & Estuary [No. A17001]. The authors wish to thank the anonymous reviewers for their constructive comments. References Altomare, C., Crespo, A.J.C., Rogers, B.D., Dominguez, J.M., Gironella, X., G� omezGesteira, M., 2014. Numerical modelling of armour block sea breakwater with smoothed particle hydrodynamics. Comput. Struct. 130, 34–45, 0. Altomare, C., Domínguez, J.M., Crespo, A.J.C., Gonz� alez-Cao, J., Suzuki, T., G� omezGesteira, M., Troch, P., 2017. Long-crested wave generation and absorption for SPHbased DualSPHysics model. Coast Eng. 127, 37–54. Altomare, C., Domínguez, J.M., Crespo, A.J.C., Suzuki, T., Caceres, I., G� omezGesteira, M., 2015. Hybridization of the wave propagation model SWASH and the meshfree particle method SPH for real coastal applications. Coast Eng. J. 57 (4), 1550024. Altomare, C., Suzuki, T., Dominguez, J.M., Crespo, A.J., Gomez-Gesteira, M., Caceres, I., 2014. A hybrid numerical model for coastal engineering problems. Coast. Eng. Proc. 1 (34), 60. Antuono, M., Colagrossi, A., Marrone, S., Molteni, D., 2010. Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Comput. Phys. Commun. 181 (3), 532–549. Barcarolo, D.A., Le Touz� e, D., Oger, G., de Vuyst, F., 2014. Adaptive particle refinement and derefinement applied to the smoothed particle hydrodynamics method. J. Comput. Phys. 273, 640–657, 0. Buchner, B., 2002. Green Water on Ship-type Offshore Structures. Crespo, A.J.C., Domínguez, J.M., Rogers, B.D., G� omez-Gesteira, M., Longshaw, S., Canelas, R., Vacondio, R., Barreiro, A., García-Feal, O., 2015. DualSPHysics: opensource parallel CFD solver based on smoothed particle hydrodynamics (SPH). Comput. Phys. Commun. 187, 204–216. Dalrymple, R.A., Rogers, B.D., 2006. Numerical modeling of water waves with the SPH method. Coast Eng. 53 (2–3SI), 141–147. De Leffe, M., Le Touz�e, D., Alessandrini, B., 2010. SPH modeling of shallow-water coastal flows. J. Hydraul. Res. 48 (SI), 118–125. Domínguez, J.M., Crespo, A.J.C., Valdez-Balderas, D., Rogers, B.D., G� omez-Gesteira, M., 2013. New multi-GPU implementation for smoothed particle hydrodynamics on heterogeneous clusters. Comput. Phys. Commun. 184 (8), 1848–1860. Engsig-Karup, A.P., Bingham, H.B., Lindberg, O., 2009. An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys. 228 (6), 2100–2118. Federico, I., Marrone, S., Colagrossi, A., Aristodemo, F., Antuono, M., 2012. Simulating 2D open-channel flows through an SPH model. Eur. J. Mech. B Fluid 34, 35–46, 0. Ferrand, M., Joly, A., Kassiotis, C., Violeau, D., Leroy, A., Morel, F., Rogers, B.D., 2017. Unsteady open boundaries for SPH using semi-analytical conditions and riemann solver in 2D. Comput. Phys. Commun. 210, 29–44. Ferrand, M., Laurence, D.R., Rogers, B.D., Violeau, D., Kassiotis, C., 2013. Unified semianalytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method. Int. J. Numer. Methods Fluids 71 (4), 446–472. Fourtakas, G., Dominguez, J.M., Vacondio, R., Rogers, B.D., 2019. Local uniform stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models. Comput. Fluids 190, 346–361. Fourtakas, G., Stansby, P.K., Rogers, B.D., Lind, S.J., Yan, S., Ma, Q.W., 2017. On the coupling of incompressible sph with a finite element potential flow solver for nonlinear free surface flows. In: The 27th International Ocean and Polar Engineering Conference. Gingold, R.A., Monaghan, J.J., 1977. Smoothed particle hydrodynamics - theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181, 375–389. Gomez-Gesteira, M., Rogers, B.D., Crespo, A.J.C., Dalrymple, R.A., Narayanaswamy, M., Dominguez, J.M., 2012. SPHysics - development of a free-surface fluid solver - Part 1: theory and formulations. Comput. Geosci. Uk 48, 289–299, 0. Gotoh, H., Khayyer, A., Ikari, H., Arikawa, T., Shimosako, K., 2014. On enhancement of incompressible SPH method for simulation of violent sloshing flows. Appl. Ocean Res. 46, 104–115. Gotoh, H., Khayyer, A., 2018. On the state-of-the-art of particle methods for coastal and ocean engineering. Coast Eng. J. 60 (1), 79–103. Gotoh, H., Shao, S.D., Memita, T., 2004. SPH-LES model for numerical investigation of wave interaction with partially immersed breakwater. Coast Eng. J. 46 (1), 39–63. Gui, Q., Dong, P., Shao, S., 2015. Numerical study of PPE source term errors in the incompressible SPH models. Int. J. Numer. Methods Fluids 77 (6), 358–379. Kassiotis, C., Ferrand, M., Violeau, D., Rogers, B., Stansby, P., Benoit, M., 2011. Coupling SPH with a 1-D boussinesq-type wave model. In: 6th International SPHERIC Workshop.

5. Conclusions This work proposes a two-way open boundary coupling scheme and a modified one-way open boundary coupling scheme based on a non­ reflective FOBC to hybridize a SWE-SPH model with a NS-SPH model. The NS-SPH model is used to present the nonlinear hydrodynamic fea­ tures in the concerned areas, while the rest of the areas are simulated with the SWE-SPH model, which calculates much faster than the former model. The hybridization performance on the information transmitting, mass and energy conservation, simulation detail, computational accu­ racy, computational efficiency and computational convergence speed is evaluated and discussed using five test cases, including solitary wave propagations, steady flows over a bump, dam breaking waves runup on a slope, dam breaking waves impacting against a vertical wall and standing waves. It is indicated that the proposed open boundary coupling schemes are capable of improving the computational efficiency on the premise of computational accuracy and simulation detail. The information transmitting and the conservation of system mass are both satisfactory. It should be noted that the open boundary condition in this paper is implemented based on the buffer zone techniques, which basically as­ sume that there is only one water surface at a given section. Thus, cases with multiple surfaces (e.g., wave breaking at the open boundary) are technically not properly treated. Numerical unstable issues may also occur when where is a strong vortex across the open boundary. More advanced open boundary techniques (Ferrand et al., 2017) for the SPH method are expected to be applied to further improve the hybridization robustness and the simulation accuracy for violent hydrodynamic con­ ditions. Additionally, well-developed grid-based models (e.g., FUN­ WAVE, SWASH and OpenWave3D, etc.), which are governed by nonlinear shallow water equations or Boussinesq equations, can also be coupled with NS-SPH using the proposed scheme and be applied to 16

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