Research Progress of SPH Simulations for Complex Multiphase Flows in Ocean Engineering

Abstract

Complex multiphase flow problems in ocean engineering have long been challenging topics. Problems such as large deformations at interfaces, multi-media interfaces, and multiple physical processes are difficult to simulate. Mesh-based algorithms could have limitations in dealing with multiphase interface capture and large interface deformations. On the contrary, the Smoothed Particle Hydrodynamics (SPH) method, as a Lagrangian meshless particle method, has some merit and flexibility in capturing multiphase interfaces and dealing with large boundary deformations. In recent years, with the improvement of SPH theory and numerical models, the SPH method has made significant advances and breakthroughs in terms of theoretical completeness and computational stability, which starts to be widely used in ocean engineering problems, including multiphase flows under atmospheric pressure, high-pressure multiphase flows, phase-change multiphase flows, granular multiphase flows and so on. In this paper, we review the progress of SPH theory and models in multiphase flow simulations, discussing the problems and challenges faced by the method, prospecting to future research works, and aiming to provide a reference for subsequent research.

1. Introduction

Multiphase flow phenomena widely exist and have an important influence on physical processes under various flow states. In ocean engineering, multiphase flow phenomena are widely present in wave breaking, bubble rising, underwater explosions, coastal evolution, and other engineering problems [1]. In the case of multiphase fluids with excessively different densities, the large-density fluid is virtually unaffected by the small-density fluid. Therefore, the multiphase flows are dominated by the large-density fluid phase and can be simplified to a single-phase free surface problem. In such cases, the small-density fluid phase is often neglected, and only the large-density fluid phase is simulated [2,3,4,5,6,7,8].

However, there are many situations where the effect of the small-density fluid phase cannot be ignored. For example, multiphase flow phenomena must be considered in problems with wave breakage and air entrapment [9], in which some complex flow phenomena occur during the evolution of the water-gas interface, including free surface breakage, overturning, fusing, and droplet splashing. The multiphase flow significantly affects the shape of liquid surfaces and the subsequent splashing direction of liquid, which in turn changes the flow characteristics of the entire flow field to some extent [10]. For example, when a ship is sailing at high speed, the air captured by free surfaces mixes with waves to form a water-air mixture, creating a white sailing wake observed from a macroscopic point of view [11]. Underwater explosions will produce high-pressure bubbles. In bubble dynamics and underwater explosion problems, multiphase flow effects are even more non-negligible [12,13,14]. The shock waves generated by the collapse of high-pressure bubbles can significantly damage ocean structures [15,16]. Besides gas-liquid multiphase flows, solid-liquid multiphase flows are not negligible in ocean engineering. A typical solid-liquid multiphase flow is the water-sand multiphase flow, which is widely found in practical engineering problems such as harbor dredging, reclamation, and coastal riverbed evolution [17]. Besides, multiphase flows in ocean engineering also contain phase-change multiphase flows, which mainly exist in cavitation and solidification phenomena. Therefore, phase-change multiphase flows cannot be ignored in ocean engineering. The engineering problems listed above show that the simulation of multiphase flows is important in ocean engineering. However, multiphase flow problems generally have complex characteristics such as large deformations, multi-media, and even phase change. These problems are very difficult to solve using traditional theoretical, experimental, or numerical simulation methods. Therefore, multiphase flow problems have become a popular and challenging research area in recent years.

Experimental studies and numerical simulations are becoming two of the most important tools used by researchers. Experimental studies provide the basis for the validation of numerical simulations. Numerical simulations can significantly reduce the number of tests, shorten the research cycle and save costs. In addition to directly guiding engineering design, the experimental study is one of the best ways to check the results of theoretical or numerical simulations. There are various experimental research results for multiphase flow problems. The Rome INSEAN towing tank laboratory gave a series of experimental results based on the Athena model in the problem of breaking waves at the bow of high-speed ships, which become a classic example of numerical results for verifying breaking waves at the bow [18]. As a classical problem in fluid mechanics, many articles related to bubble dynamics have been published [15,19], and they focus on bubble morphology, bubble fusion, bubble splitting, and bubble break-up at free liquid surfaces. In the study of high-pressure bubbles, there are numerous experimental studies on the dynamic characteristics of exploding bubbles under different boundary conditions, which can be found in the literature, such as [12,20]. Experimental studies also have their shortcomings. The main limitation is the inability to scale down all experimental conditions, creating a scaling effect. For example, scaled ship models are generally used when conducting model tests in towing tanks, but the surface tension and viscous coefficient of fluid are difficult to scale down. In addition to scale effects, experimental studies have many other limitations. For example, only a single physical quantity can be measured at some locations during tests. Due to the limited number of experimental facilities, when there are many testing cases, the tests can only be carried out in a serial manner and sequence.

In recent years, with the rapid development of computer technology, numerical simulation has gradually become an effective way to solve engineering problems. Computational Fluid Dynamics (CFD) has gradually become an important tool for hydrodynamic research in ocean engineering. However, the accurate numerical simulation of multiphase flow problems has always been one of the most challenging topics in CFD due to the complex physical phenomena. Currently, most of the numerical methods for multiphase flows are based on the Eulerian grid method. A typical Eulerian grid method is the Finite Volume Method (FVM), which adopts the Volume of Fluid (VOF) method or Level Set Method (LSM) to capture fluid interfaces [19,21]. The basic principle of these methods is the discretization of fluid domains on an Eulerian grid system, which enables the numerical calculation of hydrodynamic problems. After a long development, a relatively mature theoretical frame is gradually established and widely used in ocean engineering. However, mesh-based simulations of multiphase flows often require complex algorithms to capture or reconstruct the multiphase interface. Therefore, the stability and accuracy of interface algorithms can be a significant challenge when multiphase flows are violent. On the other hand, Lagrangian mesh methods, such as the Finite Element Method (FEM), can capture multiphase interfaces naturally. However, mesh distortions at large deformation interfaces can result in large errors and even terminate the simulation.

Lagrangian particle methods are advantageous when simulating the large deformation of fluid interfaces. The Smooth Particle Hydrodynamics (SPH) method and the Moving Particle Semi-implicit (MPS) method [22,23,24,25,26,27] are two of the main Lagrangian meshless algorithms. However, since the MPS method is based on the assumption of complete incompressibility and solves the pressure distribution implicitly based on the Pressure Poisson Equation (PPE), it is difficult to be applied to problems with significant compressible effects, such as bubble oscillations. The SPH method solves the pressure explicitly based on the Equation of State (EOS), making it ideal for solving compressible multiphase flow problems.

The SPH method was proposed by Gingold and Monaghan [28], as well as Lucy [29], which was initially used to model non-axisymmetric phenomena in astrophysics. After that, it was applied to fluid mechanics thanks to the Lagrangian property, and it becomes one of the potential methods for simulating complex fluid motions [30]. In principle, the SPH method uses a set of Lagrangian particles with physical properties to discretize fluid domains [31]. The spatial derivatives are then calculated based on the kernel approximation, which can be used in integral form for the gradient and divergence of a particular physical field, resulting in a numerical solution. SPH particles carry physical quantities such as velocity, pressure, density, and so on. Then, they move in a Lagrangian fashion according to the velocity of fluid fields. The motion of these particles represents the fluid flow, and the parameters carried by particles represent local physical quantities [32].

As a Lagrangian particle method, the SPH method strictly guarantees mass conservation. Because fluid domains are discretized into particles with physical properties, the SPH method does not require an interface tracking method for multiphase flow problems. The flowing particles represent the motion of fluid so that the motion of liquid surfaces or multiphase fluid interfaces can be determined directly. The SPH method is well suitable for simulating the multiphase flow or free surface problems and has the following main advantages:

• The SPH method is suitable for handling large deformations and fast-moving free surfaces, as well as splashing discrete droplets, which is challenging to accomplish with grid-based methods.
• SPH particles automatically capture and generate multiphase flow interfaces, which is beneficial for dealing with the coupling of multi-physics fields.
• The WCSPH method is based on the explicit solution and the independence of particle computation, which is well suitable for parallel computation, either CPU (Central Processing Unit) parallel method [33] or GPU (Graphics Processing Unit) parallel method [34].

The SPH method can accurately satisfy mass conservation during calculations and is excellent at capturing free liquid surfaces, which is beneficial for simulating complex phenomena such as the free surface break-up and droplet splashing [35,36]. However, the traditional SPH method still faces the following problems: (1) Pressure calculations are generally accompanied by severe numerical oscillations in the pressure, velocity or vorticity fields. (2) The particle resolution can not be easily adjusted dynamically. (3) Particles that are too dense or sparse in strongly compressible problems can lead to accumulated computational errors. (4) Some flow regions are prone to inducing numerical instabilities such as Tensile Instability (TI), and hence numerical voids [37,38]. Facing these problems and challenges, after half a century of development, the SPH method has been significantly optimized in terms of theory and numerical techniques, which has made great progress in computational accuracy and stability. With the development of GPU parallel technology, the computational efficiency of SPH methods has also been significantly improved, and it is capable of solving large-scale engineering problems. With the emergence and development of multi-algorithm coupling techniques embedded in SPH, the advantages of SPH methods have been fully exploited, providing a new high-precision and high-efficiency solution to multiphase flow problems with complex multi-physics fields.

In summary, in order to further elaborate the research progress of SPH theory and models in multiphase flow problems, this paper will review the basic principle of SPH methods and the progress of numerical techniques, focusing on the multiphase flows under atmospheric pressure, high-pressure multiphase flows, phase-change multiphase flows and granular multiphase flows (Figure 1). The future development of SPH methods in multiphase flow problems is also discussed to provide a reference for the research on multiphase flow problems.

2. The Basic Theory and Developments of SPH Models

After half a century of development, SPH methods have developed in various forms based on different theoretical frameworks (Figure 2). Based on the compressibility assumption of fluid, the two main types of methods are Weakly Compressible SPH (WCSPH) methods and Incompressible SPH (ISPH) methods, which have been widely used in their respective fields. Below, we review the basic principles and recent developments of SPH models.

2.1. A Brief Recall of SPH Methods

Liquids, e.g., water and oil, are generally considered to be approximately incompressible, but Monaghan et al. [39] first introduced the EOS in order to solve the pressure based on density, treating the fluid as weakly compressible. When the EOS is used to solve the pressure, the integral time step is greatly limited by the Courant Friedrichs Lewy condition (CFL condition). In short, the time step is greatly limited by the sound speed. Therefore, Monaghan et al. [40] proposed the weakly-compressible hypothesis to maintain the approximate incompressible properties of fluid by adopting the artificial sound speed with a Mach number less than 0.1, which ensures that the fluid density varies within 1%. Therefore, the method is also known as the Weakly-Compressible SPH method (WCSPH method). The use of the artificial sound speed enlarges the time step and therefore improves the efficiency of simulations.

In contrast, the density of flow fields remains strictly constant in the ISPH method. In short, the velocity divergence is zero everywhere. Rafiee et al. [41] used the PPE to solve pressure at each time step within the framework of SPH methods and proposed the ISPH method to simulate incompressible flow problems. The ISPH method is very similar to the MPS method. They both belong to the semi-implicit approach, based on the assumption of fluid incompressibility, and use the PPE [42,43,44,45,46,47,48,49,50]. However, ISPH and MPS methods involve large sparse matrix operations and free liquid surface particle detection, resulting in higher computational cost. At the same time, parallel computation is not as feasible as in WCSPH due to the implicit solution of pressures [51,52], which is more challenging for large-scale 3D computations. Although compressible source terms can be added to simulate weakly-compressible problems, the simulation of strongly-compressible problems is still challenging. The WCSPH method is based on the EOS, so it is suitable for solving compressible multiphase flows. In the past 20 years, the computational accuracy of SPH methods has been significantly improved. Compared with the ISPH method, the WCSPH method has a smaller storage capacity in the GPU parallelization [53] and can reach higher computational efficiency. Therefore, this paper mainly targets the theoretical framework of the WCSPH method and reviews the SPH simulations of multiphase flows in ocean engineering.

The Navier-Stokes (NS) equations for compressible fluids in Lagrangian formalism are written as follows [54]:

$$\{\begin{array}{l}\frac{\mathrm{D}\rho }{\mathrm{D}t}=-\rho \nabla ·\mathit{u},\\ \frac{\mathrm{D}\mathit{u}}{\mathrm{D}t}=\frac{\nabla ·\mathit{T}}{\rho }+\mathit{g},\\ \frac{\mathrm{D}\mathit{r}}{\mathrm{D}t}=\mathit{u},\\ \frac{\mathrm{D}e}{\mathrm{D}t}=\frac{\mathit{T}:\mathit{D}}{\rho }-\nabla ·\mathit{q},\\ p=f(\rho ,e).\end{array}$$

(1)

In the above equations, $D/Dt$ represents the material derivative of physical quantities, called the material derivative or Lagrangian derivative to time t. $\rho$, $\mathit{u}$, $\mathit{r}$, $\mathit{q}$, e, and p represent the density, velocity vector, position vector, heat flux, internal energy and pressure of the fluid, respectively. $\mathit{g}$ is the acceleration due to the body force (in this paper, the body force is gravity). $\mathit{T}$ and $\mathit{D}$ are the stress tensor and strain rate tensor of the Newtonian fluid, which are defined as follows [55]:

$$\mathit{T}=(-p+\lambda tr\mathit{D})\mathit{I}+2\mu \mathit{D},\mathit{D}=\frac{\nabla \mathit{u}+\nabla ·{\mathit{u}}^T}{2}.$$

(2)

The divergence of $\mathit{T}$ is shown as:

$$\nabla ·\mathit{T}=-\nabla p+(\lambda +\mu )\nabla (\nabla ·\mathit{u})+\mu {\nabla }^2\mathit{u},$$

(3)

where $\mu$ and $\lambda$ represent the dynamic and the bulk viscosity coefficients, respectively. $\mathit{I}$ is an identity matrix.

The particle approximation equations aim to discretize the spatial derivative of governing equations, which are based on the kernel approximation, and they are as follows [54]:

$$\{\begin{array}{c}f\left({\mathit{r}}_i\right)={\displaystyle \sum _j}f\left({\mathit{r}}_j\right)W\left({\mathit{r}}_i-{\mathit{r}}_j,h\right)V_j,\hfill \\ {\nabla }_{i}f\left({\mathit{r}}_i\right)={\displaystyle \sum _j}\left[f\left({\mathit{r}}_j\right)+f\left({\mathit{r}}_j\right)\right]{\nabla }_{i}W\left({\mathit{r}}_i-{\mathit{r}}_j,h\right)V_j,\hfill \\ {\nabla }_i·\mathit{f}\left({\mathit{r}}_i\right)={\displaystyle \sum _j}\left[\mathit{f}\left({\mathit{r}}_j\right)-\mathit{f}\left({\mathit{r}}_j\right)\right]·{\nabla }_{i}W\left({\mathit{r}}_i-{\mathit{r}}_j,h\right)V_j,\hfill \end{array}$$

(4)

where f and $\mathit{f}$ represent the scalar and vector, respectively. V represents the volume of particles. The subscripts i and j represent the target particle and the neighboring particle, respectively. W is the kernel function, and h is the smooth length.

When the viscous force and internal energy are not considered, based on the particle approximation equations described above, the governing equations for a barotropic fluid of the traditional WCSPH method are discretized as follows [54]:

$$\{\begin{array}{c}\frac{D{\rho }_i}{Dt}=-{\rho }_i{\displaystyle \sum _j}\left({\mathit{u}}_j-{\mathit{u}}_i\right)·{\nabla }_i{W}_{ij}{V}_j,\hfill \\ \frac{D{\mathit{u}}_i}{Dt}=-\frac{1}{{\rho }_i}{\displaystyle \sum _j}\left(p_i+p_j\right){\nabla }_i{W}_{ij}{V}_j+\mathit{g},\hfill \\ \frac{D{\mathit{r}}_i}{Dt}={\mathit{u}}_i,\hfill \\ p_i=c_0^2\left(\rho -{\rho }_0\right),\hfill \end{array}$$

(5)

where $c_0$ and ${\rho }_0$ are the sound speed and reference fluid density, respectively.

One of the key challenges of SPH methods is efficiently performing accurate and stable simulations. The smoothing kernel function used for kernel and particle approximation affects the accuracy. Many researchers have studied smoothing kernel functions, and various kernel functions have been proposed [56]. Recently, three of the most widely used kernel functions are the cubic spline, quintic spline, Gaussian, and Wendland kernel functions.

2.2. Improved SPH Models

To improve the accuracy and stability of SPH methods, different improved SPH models are established. To verify the simulation of improved SPH models for multiphase flows, researchers have proposed different benchmarks listed in

Table 1. Benchmark tests of SPH simulations for multiphase flows.

Benchmark Tests	Comparison Results	References
Oscillating drop under a central force field	Analytic solutions	Monaghan et al. [57] Khayyer et al. [58] Hammani et al. [59]
2-D SPH Validation: Effect of wet bottom on dam break evolution	Experimental results	M Jánosi et al. [60] Crespo et al. [61] Yang et al. [62]
Sloshing with air entrapment	Experimental results	Souto-Iglesias et al. [63,64]
Wave impact with air entrapment	Experimental results	Luo et al. [65] Sun et al. [66]
Bubbles rising and coelleasing	Experimental results	Bhaga et al. [67] Brereton et al. [68] Zhang et al. [14]
Underwater explosion	Experimental results	Li, Cui, Ming et al. [12,15,16] Sun et al. [69,70]
Cavitating flows	Experimental results	Huang et al. [71] Müller et al. [72] Lyu et al. [73]
Icing process	Experimental results	Šikalo et al. [74] Cui et al. [75]
Granular multiphase flows	Experimental results	Sun et al. [76] Ghaitanellis et al. [77] Xie et al. [78]

. In the following section, we introduce some typical improved SPH variants.

2.2.1. $\delta$-SPH Mdel

In the ISPH and MPS methods, a smoother pressure field can naturally be obtained due to the incompressibility assumption and the implicit solution of pressures using the PPE [10,26,79,80,81]. However, in the early WCSPH methods, significant pressure field fluctuation was evident and difficult to avoid due to the weak compressibility assumption. The errors in the approximation of the SPH kernel function and numerical discretization lead to fluctuations in the density field of particles, and the pressure of particles is calculated from the EOS so that small changes in density can lead to violent pressure field fluctuations [82]. Because the actual sound speed increases the calculation time and does not significantly improve the accuracy of calculations, an artificial sound speed smaller than the actual sound speed is generally used to reduce the time step and improve the efficiency of the calculation. When the fluid is sloshing and thumping, the pressure wave at the thumping location propagates through the flow field at a low sound speed. The pressure waves are reflected and superimposed several times when they reach the boundary, again leading to pressure field fluctuations [59].

In order to reduce the pressure fluctuations in WCSPH simulations, researchers have proposed many models. The foundational work can be traced back to Monaghan and Gingold, who proposed an artificial viscosity term based on the Neumann-Richtmeyer viscosity to reduce shock wave-induced oscillations [39]. The next breakthrough was the SPH density filtering technique, which was proposed by Colagrossi et al. [83] and can reduce the pressure fluctuations in SPH simulations. Although the method can provide better simulation results, it cannot be applied to long-time simulations. Because the hydrostatic pressure component of fluid is non-physically filtered, it can lead to a non-conservation of the fluid volume.

In 2010, Antuono et al. [84] proposed the $\delta$-SPH model, which can obtain good density/pressure solutions in most cases. Marrone et al. [85] applied the $\delta$-SPH model to the flow with free liquid surfaces. The $\delta$-SPH model introduces a density diffusive term that acts similarly to the Riemann-Solver diffusive term, which can improve the accuracy of the pressure field.

The governing equations for the $\delta$-SPH model are as follows [86]:

$$\{\begin{array}{c}\frac{D{\rho }_i}{Dt}=-{\rho }_i{\displaystyle \sum _j}\left({\mathit{u}}_j-{\mathit{u}}_i\right)·{\nabla }_i{W}_{ij}{V}_j+\delta h{c}_0{D}_i,\hfill \\ \frac{D{\mathit{u}}_i}{Dt}=-\frac{1}{{\rho }_i}{\displaystyle \sum _j}\left(p_i+p_j\right){\nabla }_i{W}_{ij}{V}_j+\mathit{g}+\frac{{\mathit{F}}_i^V}{{\rho }_i},\hfill \\ \frac{D{\mathit{r}}_i}{Dt}={\mathit{u}}_i,p=c_0^2\left(\rho -{\rho }_0\right),\hfill \\ D_i=2{\displaystyle \sum _j}{\psi }_{ij}\left({\mathit{r}}_j-{\mathit{r}}_i\right)·{\nabla }_i{W}_{ij}{V}_j/{\left|{\mathit{r}}_j-{\mathit{r}}_i\right|}^2,\hfill \\ {\psi }_{ij}=\left({\rho }_j-{\rho }_i\right)-0.5\left[{\langle \nabla \rho \rangle }_i^L+{\langle \nabla \rho \rangle }_j^L\right]·\left({\mathit{r}}_j-{\mathit{r}}_i\right).\hfill \end{array}$$

(6)

The last term in the continuity equation is the density diffusive term. $\delta$ is generally set as 0.1. ${\mathit{F}}_i^V$ is the viscous force. The physical viscous force can be implemented as follows [87]:

$${\mathit{F}}_i^V=\{\begin{array}{ll}(K\mu +\alpha h{c}_0{\rho }_0){\displaystyle \sum _j}{\pi }_{ij}{\nabla }_i{W}_{ij}{V}_j\hfill & i\mathrm{or}j\mathrm{belong}\mathrm{to}\mathrm{free}\mathrm{surface},\\ K\mu {\displaystyle \sum _j}{\pi }_{ij}{\nabla }_i{W}_{ij}{V}_j\hfill & \mathrm{else}.\end{array}$$

(7)

In the Equation (7), ${\pi }_{ij}$ is defined as follows [87]:

$${\pi }_{ij}=\left({\mathit{u}}_j-{\mathit{u}}_i\right)·\left({\mathit{r}}_j-{\mathit{r}}_i\right)/{\left|{\mathit{r}}_j-{\mathit{r}}_i\right|}^2.$$

(8)

In order to avoid numerical instability caused by the breaking of free surfaces, the particles at free surfaces are applied with an artificial viscosity on the basis of physical viscosity. In the physical viscosity term, $K=2(d+2)$, and d is the dimension.

Besides, $c_0$ is the artificial sound speed, which is chosen by $c_0\ge 10max(U_{max},\sqrt{\frac{p_{max}}{{\rho }_0}})$. The $U_{max}$ and $p_{max}$ are the estimated maximum velocity and pressure in the flow.

For the simulations of inviscid flows, only artificial viscosity is used. $\alpha$ uses a smaller value (0.01–0.02) for low-speed hydrodynamic problems but a larger value (about 0.1–0.5) for shock or impact problems. The other physical quantities in the equations are defined similarly as in Equations (1), (4) and (5).

Despite the success of the $\delta$-SPH model, there are still some unresolved problems in this SPH model, which contain TI and non-uniform spatial configuration of fluid particles. For example, in the simulation of flows around an airfoil, TI can lead to the development of negative pressure regions in the tail region and induce non-physical cavity on the airfoil surface, which can make the calculations of $\delta$-SPH model inaccurate [88]. Another problem arises from the Lagrangian property of SPH methods, which leads to a non-uniform spatial configuration of fluid particles in a flow region. In addition, there are numerical fluctuations in the vorticity field of $\delta$-SPH results when simulating the viscous flow problems considering boundary layers, so Sun and co-workers [82] proposed the ${\delta }^{+}$-SPH model to overcome the numerical instability problem.

2.2.2. ${\delta }^{+}$-SPH Model

Nestor et al. [89] were the first to propose a technique in the Finite Volume Particle Method (FVPM), which allows for a small arbitrary Lagrange-Euler correction to the motion of Lagrangian particles. The algorithmic accuracy of SPH methods relies strongly on the uniformity of particles, so Lind et al. [90] proposed the Particle Shifting Technique (PST) inspired by the work of Nestor et al. [89], which initially can solve the problems of particle non-uniformity in ISPH simulations. After that, the PST was improved by Khayyer et al. [91] in the simulations of ISPH. To overcome the non-uniform spatial configuration of fluid particles in the WCSPH method, Sun et al. [82,92] combined the PST with the $\delta$-SPH model and proposed the ${\delta }^{+}$-SPH model. At the same time, Sun et al. [82] proposed a correction for the displacement of particles at free liquid surfaces to guarantee the kinematic boundary. The principle of this correction is that the shifting vector in the normal direction to the free surface is made zero, and only the shifting in the tangential direction is retained. This treatment ensures that the particles remain uniformly distributed at the free surface. Subsequently, Sun et al. [93] further developed the ${\delta }^{+}$-SPH model that satisfies the consistency by expressing the particle shifting as a modified velocity and taking it into account in the governing equations. It solves the problem of non-physical volume expansion in free surface flows and greatly improves the accuracy and stability of WCSPH methods.

The problem of TI was a unique numerical inherent deficiency in SPH methods. Swegle et al. [94] conducted an in-depth discussion on TI from a theoretical perspective for the first time. He pointed out that TI is related to the stress state of particles and independent of the magnitude of stress. Lyu et al. [38] further proved that TI is the inherent deficiency of SPH methods, which is caused by the particle dispersion error of differential operators. The traditional numerical instability should gradually converge to zero with the refinement of particle resolution, but TI does not have this feature, which means that special numerical techniques are required to suppress TI. As a result, Lyu et al. [38] pointed out that TI cannot be regarded as a kind of numerical instability. Researchers found that the PST can partly suppress TI to a certain extent, but it is not enough to eliminate it [82]. In order to overcome this inherent deficiency, Sun et al. [37] proposed the Tensile Instability Control (TIC) technique, which overcomes the numerical cavity in the high negative pressure region and is successfully applied to the simulation of viscous boundary layers with medium and high Reynolds numbers [88,95].

The main difference between the ${\delta }^{+}$-SPH model and the traditional $\delta$-SPH model is the equation related to the motion of fluid particles. The ${\delta }^{+}$-SPH model requires an additional correction to ensure the uniformity of particles, which is based on the position update of particles in the ${\delta }^{+}$-SPH model. Finally, the equations of the particle shifting are as follows [82]:

$$\{\begin{array}{c}\frac{D{\mathit{r}}_i}{Dt}={\mathit{u}}_i,\hfill \\ {\mathit{r}}_i^{*}=\left({\mathit{r}}_i+\delta {\mathit{r}}_i\right),\hfill \\ \delta {\mathit{r}}_i=-CFL·\mathrm{Ma}·{\left(2h_i\right)}^2·{\displaystyle \sum _j}\left[1+R{\left(\frac{W_{ij}}{W\left(\Delta x_i\right)}\right)}^n\right]{\nabla }_i{W}_{ij}\frac{m_j}{\left({\rho }_i+{\rho }_j\right)},\hfill \end{array}$$

(9)

where $\mathrm{Ma}$ is the Mach number, and m represents the mass of particles. The other physical quantities in the equations are defined similarly as in Equations (1), (4) and (5).

Recently, Antuono et al. [96] proposed the $\delta$-ALE-SPH model in Arbitrary Lagrangian-Eulerian (ALE) form, which introduced the mass equation and the numerical dissipation term into governing equations. It was also proved that, for weakly-compressible simulations, the $\delta$-ALE-SPH model [96] and the consistent ${\delta }^{+}$-SPH model [93] practically give very similar results.

2.2.3. SPH Model Based on Riemann Solvers

As a variant of SPH methods, the SPH model based on Riemann solvers has been widely used [97,98,99,100]. The SPH model based on Riemann solvers introduces implicit numerical dissipation other than using explicit artificial viscosity and achieves the dissipation in a more accurate manner.

The fundamental governing equations for the SPH model based on Riemann solvers are written as follows [101]:

$$\{\begin{array}{c}\frac{D{\rho }_i}{Dt}=2{\rho }_i{\displaystyle \sum _j}\left({\mathit{u}}_i-{\mathit{u}}^{*}\right)·{\nabla }_i{W}_{ij}{V}_j,\hfill \\ \frac{D{\mathit{u}}_i}{Dt}=-2{\displaystyle \sum _j}\frac{p^{*}}{{\rho }_i}{\nabla }_i{W}_{ij}{V}_j+\mathit{g},\hfill \\ \frac{D{e}_i}{Dt}=\frac{2}{{\rho }_i}{\displaystyle \sum _j}{p}^{*}\left({\mathit{u}}_i-{\mathit{u}}^{*}\right)·{\nabla }_i{W}_{ij}{V}_j,\hfill \\ u^{*}=\frac{1}{2}\left[u_L+u_R+\frac{1}{S}\left(p_L-p_R\right)\right],p^{*}=\frac{1}{2}\left[p_L+p_R+S\left(u_L-u_R\right)\right],\hfill \\ S=\frac{c_R{\rho }_R\sqrt{{\rho }_R}+c_L{\rho }_L\sqrt{{\rho }_L}}{\sqrt{{\rho }_R}+\sqrt{{\rho }_L}}.\hfill \end{array}$$

(10)

The left and right states in SPH are written as follows [101]:

$$\{\begin{array}{c}{f}_L=\left({\rho }_L,u_L,p_L\right)=\left({\rho }_i,{\mathit{u}}_i·{\mathit{e}}_{ij},p_i\right),\hfill \\ f_R=\left({\rho }_R,u_R,p_R\right)=\left({\rho }_j,{\mathit{u}}_j·{\mathit{e}}_{ij},p_j\right),\hfill \\ e_{ij}=-\frac{x_{ij}}{\left|x_{ij}\right|},\hfill \end{array}$$

(11)

where the initial values carried by particles i and j are defined as the left state $f_L$ and the right state $f_R$, and S is the average Lagrangian speed of sound between left and right states. The other physical quantities in equations are defined similarly as in Equations (1), (4) and (5).

Vila [102] first proposed an SPH model based on Riemann solvers. This model can rapidly dissipate the pressure pulsations caused by SPH numerical errors and weak compressibility assumptions, which can improve the smoothness of pressure fields. The model adopts an arbitrary Lagrangian-Eulerian description, which differs from the standard SPH that uses a pure Lagrangian description. The form of this model is similar to the FVM, which uses a control volume with arbitrary velocity to solve a conservative form of the Euler equation. In addition, it adopts Riemann solvers to calculate the flux on a moving control volume rather than particles. Parshikov et al. [103] proposed an SPH formulation based on Riemann solvers in a pure Lagrangian framework by using a first-order Riemann solution to describe the contact interaction between particles. Inutsuka [104] reformulated a Riemann-based SPH scheme with a 2nd-order Riemann solution. The scheme introduces the minimum and sufficient amount of dissipation in the equations by using an iterative solution to a non-linear Riemann problem at an imaginary interface between an interacting particle pair. The scheme was named the Godunov SPH (GSPH) and is attractive because it eliminates parametrization and user intervention, typically associated with the SPH artificial viscosity. The governing equations of the SPH model based on Riemann solvers have been reformulated on the foundation of standard SPH equations. The SPH model based on Riemann solvers is executed by introducing the Riemann-Solver directly into the governing equations without any other operations.

2.2.4. SPH Model with Summation-Based Density

Based on the work of Espanol and Revenga [105], Hu and Adams [106] established a multiphase SPH model with summation-based density. The volume $V_i$ of particle i can be derived from the surrounding particle j with a volume interpolation using the Shepard kernel function. The neighboring particles only contribute to the volume of the particle i, but not the density, so the model can naturally deal with density discontinuities at phase interfaces. The momentum and volume equations are as follows [106]:

$$\{\begin{array}{c}\frac{\mathrm{D}{\mathit{u}}_i}{\mathrm{Dt}}=\frac{1}{{\rho }_i}\left(-\frac{1}{V_i}{\displaystyle \sum _j}\left(p_i{V}_i^2+p_j{V}_j^2\right){\nabla }_i{W}_{ij}+{\langle \nabla ·\mathit{V}\rangle }_i+{\mathit{F}}_i^S\right)+\mathit{g},\hfill \\ {\langle V\rangle }_i={\displaystyle \sum _j}{\chi }_{ij}{V}_j={\displaystyle \sum _j}\frac{W_{ij}}{\mathrm{\Gamma }\left({\mathit{r}}_i\right)}{V}_j=\frac{{\sum }_j{W}_{ij}{V}_j}{\mathrm{\Gamma }\left({\mathit{r}}_i\right)}\approx \frac{1}{\mathrm{\Gamma }\left({\mathit{r}}_i\right)},\hfill \end{array}$$

(12)

where ${\chi }_{ij}$ is the Shepard kernel function, and $\mathrm{\Gamma }\left({\mathit{r}}_i\right)={\sum }_j{W}_{ij}$. The ${\mathit{F}}_i^S$ is the surface tension. The other physical quantities in the equations are defined similarly, as in Equations (1), (4) and (5).

With the algorithm proposed by Hu and Adams [106], several aspects of multiphase interactions are addressed. Newly formulated viscous terms allow for a viscosity discontinuity, which ensures the continuity of velocity and shear stress across phase interfaces. A new simple algorithm capable of three or more immiscible phases is developed. Mesocopicinterface slippage is included based on the apparent slip assumption, which ensures the continuity at phase interfaces. The simulations of capillary waves, three-phase interactions, drop deformation in a shear flow, and mesoscopic channel flows were also carried out to verify the validity of this model for multiphase flow simulations [107].

3. Research Progress for Multiphase SPH Models and Numerical Techniques

3.1. Surface Tension Calculation

The surface tension present at the interface between two phases is an important characteristic of multiphase flows, especially for gas-liquid mixing flows, microflows, and surface flows. It is the main factor influencing flow behaviors. The accurate numerical representation of surface tensions is essential for multiphase flow simulations. For small-scale problems, such as rising bubbles in water, the shape of bubbles is mainly maintained by the surface tension, so the accurate application of surface tension models is critical in accurately predicting the final shape of bubbles [14]. The correct calculation of surface tensions in multiphase flows requires introducing an accurate surface tension model. Therefore, in the existing studies, the surface tension is generally calculated according to the classical Continuum Surface Force (CSF) model, which converts the surface tension from a linear force to a body force on the interfacial particles. The CSF model was first proposed by Brackbill et al. [108] and is a macroscopic force model. The surface tension is described as a biased molecular interaction between two phases, and the interface between two fluid phases is considered a transitional region of finite thickness. The magnitude of surface tension is proportional to the local curvature of surfaces. It is applied within the radius of the kernel function and distributed on both sides of interfaces according to a specific weighting function (Figure 3). The direction of surface tensions is opposite to the normal direction at interfaces, which can be obtained from the color function.

Based on the CSF model, Adami et al. [109] and Duan et al. [110] proposed a curvature calculation based on color function contours and named it the CCSF (Contoured Continuum Surface Force) model. The model starts by smoothing the discontinuous color function, then obtains the smooth color function by Taylor expansion. Finally, it calculates the curvature directly from the contour equation. As the CCSF model uses an analytical curvature solution, the accuracy of curvature calculations is improved, and an accurate calculation of the surface tension is achieved [111].

3.2. Techniques to Improve Interface Continuity and Sharpness

In multiphase flow problems, the large density and viscous ratios at water-gas interfaces are an issue of great interest to researchers. In the traditional WCSPH method, numerical instability problems in the simulation of violent multiphase flows can become very serious due to the large density discontinuities at interfaces. Colagorossi and Landrini [83] first proposed various improvement techniques. They used proper discretizations for the continuity and momentum equations, which are stable even at the density discontinuities of interfaces. However, in their SPH model, the fluid phase with small density should be assigned a much larger artificial sound speed (depends on the density ratio) than the denser phase to ensure the stability of calculations, which reduces the time step and therefore makes the simulation time longer. Recently, He et al. [112] proposed a stable SPH model with large CFL numbers for multiphase flows with large density ratios.

Colagorossi and Landrini [83] used the density filter to stabilize the pressure field. However, the density filter is conducted within each fluid phase without considering the pressure on the other side of interfaces. Chen et al. [113] proposed a density filter technique considering the effect of different fluid phases to improve the pressure continuity at the interface. In addition, they proposed an improved model using a pressure gradient operator to apply an induced particle acceleration at interfaces. Its calculation is much more efficient due to using the same sound speed for large-density and small-density fluid. In order to prevent the penetration of particles at interfaces, interface sharpness forces are added to the particles near the interface by Grenier et al. [114]. The forces are designed to be of equal magnitude but in the opposite direction, ensuring the momentum conservation.

3.3. Phase-Change Model

The typical phase-change processes in ocean engineering are cavitation and icing, which both requires the development of reasonable phase-change models. Currently, two main cavitation models are widely used: models based on mass transport and models based on the EOS. The models based on the EOS are suitable for the particle method. In the cavitation models based on the EOS for positive pressure fluid, the thermodynamic effect is generally ignored [115].

Delannoy et al. [116] originally proposed the EOS model for positive pressure fluid. In this model, the density of gas-liquid mixtures can be described by the EOS, which is considered a function of pressure and density. Usually, the effect of temperature can be neglected in the Delannoy EOS model, and the density of mixtures can be reduced to a single value function of the local pressure, as shown in Figure 4. In this model, when the pressure is larger than $p_v^{(+)}$, the mixture is considered pure liquid, and its density as a function of pressure follows the Tait equation. When the pressure is smaller than $p_v^{(-)}$, the local flow medium is considered pure vapor, and the relationship between fluid density and pressure satisfies the ideal gas EOS. When the pressure is moderate, the local flow field consists of a mixture of vapor and liquid, and a sine curve describes the density-pressure relationship. The model can better simulate the stable attached cavity. At the same time, the prediction of pressures and other parameters are in good agreement with the experimental results [117].

Lyu et al. [73] proposed a phase-change SPH model to describe a complex fluid system with multiple components based on the Delannoy EOS model, assuming that the cavitation system consists of three components, pure liquid, pure vapor, and vapor-liquid mixture. In such a system, the fluid density of flow fields is calculated by the law of positive pressure states, which is a single-valued function between density and pressure.

The Delannoy EOS is shown below [73]:

$$p_i=\{\begin{array}{ll}{c}_l^2\left({\rho }_i-{\rho }_l^{ref}\right)\hfill & {\rho }_i>{\rho }_{m^{+}},\\ \frac{c_{min}^2{\rho }_m^{(-)}}{2}arcsin\left[\frac{2{\rho }_i-{\rho }_m^{(+)}}{{\rho }_m^{(-)}}\right]+p_v\hfill & {\rho }_{m^{+}}\le p_i\le {\rho }_{m^{+}},\\ B_v\left[{\left(\frac{{\rho }_i}{{\rho }_v^{ref}}\right)}^{{\gamma }_v}-1\right]\hfill & {\rho }_i<{\rho }_{m^{-}}.\end{array}$$

(13)

In the Equation (13), ${\rho }_m^{(\pm )}$ is defined, as follows [73]:

$${\rho }_m^{(\pm )}:={\rho }_{m^{+}}\pm {\rho }_{m^{-}},{\rho }_{m^{+}}={\rho }_l^{ref}+p_v^{(+)}{c}_l^{-2},{\rho }_{m^{-}}={\rho }_v^{ref}{\left[p_v^{(-)}{B}_v^{-1}+1\right]}^{{\gamma }_v^{-1}}.$$

(14)

In the Equations (13) and (14), $p_v^{(\pm )}$ and $B_v$ are defined, as follows [73]:

$$p_v^{(\pm )}:=p_v\pm 0.5\Delta p_v,\Delta p_v=\frac{{\rho }_m^{(+)}}{2}\pi c_{min}^2{B}_v=\frac{c_v^2{\rho }_v^{ref}}{{\gamma }_v},$$

(15)

where $p_v$ represents the saturation pressure, and its value depends on the ambient pressure and temperature. ${\rho }_{m^{+}}$ and ${\rho }_{m^{-}}$ represent the upper and lower density limits of water–gas mixtures. $p_{m^{+}}$ and $p_{m^{-}}$ represent the upper and lower pressure limits of water–gas mixtures. $c_l$ and $c_v$ are the sound speed of liquid and gas. $c_{min}$ remains a tunable parameter of the model, which can be regarded as the minimum sound speed in the mixture. $p_l^{ref}$ and $p_v^{ref}$ are the reference pressure of liquid and gas. ${\rho }_l^{ref}$ and ${\rho }_v^{ref}$ are the reference density of liquid and gas. ${\gamma }_v=7$ and ${\gamma }_v=1.4$ denote the polytropic constants for pure water and vapor. The other physical quantities in the equations are defined similarly as in Equations (1), (4) and (5).

The sound speeds of liquid and gas are determined according to the following equation [73].

$$\{\begin{array}{l}{c}_l\ge 10max\left(U_{max},\sqrt{\frac{p_{max}}{{\rho }_l^{ref}}}\right),\hfill \\ c_v=\sqrt{\frac{c_l^2{\gamma }_v{\rho }_l^{ref}}{{\gamma }_l{\rho }_v^{ref}}}.\hfill \end{array}$$

(16)

In order to achieve the simulation of the ice-water phase-change process, the transformation of particles between the two phases is controlled by a latent heat storage model [75]. Specifically, when the temperature of the water particle reaches the freezing point ($T_m=273.15\mathrm{K}$), it enters the solidification phase, where the particle temperature remains constant ($T=T_m$). The heat loss in the solidification phase is recorded as $Q_{loss}\left(t\right)$. When the $Q_{loss}\left(t\right)$ in this phase is equal to the latent heat energy of the water particle ($Q_l$), the water particle is transformed into an ice particle (Figure 5).

3.4. Turbulence Models in SPH Framework

Turbulence plays a significant role in the motion of fluid, so turbulence models also play a very important role in the simulations of hydrodynamics. Recently, an increasing number of studies have been dedicated to extending SPH methods to modeling turbulent flows [118,119]. However, the modeling of turbulent flows has been a great challenge in terms of both theoretical and numerical issues. In fact, the Direct Numerical Simulation (DNS) of high Reynolds number flows remains infeasible because of the wide range of scales to be resolved. An alternative approach is the solution of the Navier-Stokes equations in the time- or ensemble-averaging formulation given by Reynolds Averaged Navier-Stokes (RANS) equations, where all the space and time turbulent scales are modeled. As large eddy evolution strongly depends on the boundary conditions and size of the domain, universal modeling for RANS is extremely difficult [120]. In the SPH simulations of turbulence, some researchers attempted to use RANS approaches, which rely on a direct inclusion of $k-ϵ$ models [121,122].

The approach that lies in the middle between DNS and RANS is Large Eddy Simulation (LES), where only sub-grid turbulent eddies are modeled by space filtering. In contrast, the largest eddies are directly simulated [123]. Because when the discretization is fine enough, the small sub-grid eddies should be closer to an “isotropic and homogeneous” state and much less dependent on the specific problem under investigation, LES models are expected to be easier than RANS models. The LES model is a good balance between computational accuracy and efficiency. A preliminary effort in such a direction was made in the work of Di Mascio et al. [124]. They proposed a Lagrangian LES-SPH scheme named $\delta$-LES-SPH, which was obtained by applying a Lagrangian filter to the NS equations for compressible fluids and approximating the differential operators in an SPH model. This model is similar to a general weakly compressible SPH scheme, apart from the presence of additional terms in both the continuity and momentum equations coming from the LES filtering procedure. The model performs well for moderate Reynolds number flows. Based on the work of Di Mascio et al. [124], Antuono et al. [125] proposed an improved $\delta$-LES-SPH model. The model can simulate high Reynolds number flows without the phenomenon of TI, which is achieved by rearranging the LES equations in the SPH formalism and combining the $\delta$- LES-SPH model with the TIC technique.

3.5. SPH Models for Simulating Granular Multiphase Flows

In ocean engineering, the granular multiphase flows usually refer to the water-sand two-phase flows. In the simulation of water-sand two-phase flows, the key issues are the model of water-phase turbulence, the effect of sediment concentration on water-phase turbulence, the choice of forces between two phases, the pulsation characteristics of solid sediment particles in turbulent flows, and the calculation of interactions between sediment particles [126,127,128]. These fundamental problems are closely related to the action of water-sand two-phase flows. They are also the problems that need to be addressed in the current water-sand two-phase flow modeling.

At present, four main models are based on the SPH method to simulate water-sand two-phase flows. The first model uses the SPH method coupled with the Discrete Element Method (DEM) to simulate water-sand two-phase flows. The SPH method simulates the fluid phase, and the DEM method simulates the sediment particles [129,130,131]. The coupled SPH-DEM method can directly calculate the movement of sediment particles. The second model considers the mixture of sediment and pore-water phase as a non-Newtonian fluid, which is treated as a phase. The fluid is then considered pure water, which is treated as another phase. This model can simulate the changes at interfaces between bed-load sediment and clear water, similar to an oil–water mixture [126,132,133]. However, the model ignores the variation of water in pores within bed-load sediment and the presence of suspended sediment. A third model uses two sets of particles to simulate mixed water–sand flows, calculating the solid and fluid phases separately [134]. In this model, it is assumed that at any given time, any point in space can be occupied by both solid and fluid particles. These two sets of particles can be superimposed in space, and their interactions are considered through interphase forces such as drag and lift. This model can capture changes in bed-load sediment and calculate fluid motion in the pore space. However, it can still not simulate the suspended sediment, and the computational scale is large.

Recently, Shi et al. [17,135] proposed a newer model in which both sediment and fluid are described using a set of SPH particles. Each SPH particle has the properties of two phases and carries a sediment volume fraction. In the bed-load sediment region, the volume fraction is close to 1. In the suspended sediment region, the volume fraction is close to 0. The sediment particles are assumed to move with the velocity of flows, and the properties of solid and liquid phases carried by SPH particles are still governed by the volume-averaged continuity and momentum equations. This model allows the calculation of bed-load sediment movements for high sediment concentrations and suspended sediment movements for low sediment concentrations with relatively low computational demands.

The EOS for a mixed water-sand flow is [17,135]:

$$p=\frac{{\rho }_{f0}{c_0}^2}{\xi }\frac{{\alpha }_f{\rho }_f+{\alpha }_s{\rho }_{f0}}{{\alpha }_f{\rho }_f}\left\{{\left[\frac{{\alpha }_f{\rho }_f+{\alpha }_s{\rho }_{f0}}{{\rho }_{f0}}\right]}^{\xi }-1\right\}$$

(17)

where $\xi$ is a constant with a value of 7. $\alpha$ represents the volume fraction, and ${\rho }_{f0}$ represents the initial density of fluid. The subscripts f and s represent the fluid and sediment particle phases, respectively. The other physical quantities in the equations are defined similarly as in Equations (1), (4) and (5).

3.6. Multi-Resolution Techniques

When the SPH method is used for fluid–solid interaction problems in multiphase flows, a single spatial resolution is usually used to discretize the fluid and solid computational domains. Specifically, uniform particle spacing is used throughout the computational domain. A single time step is used for the time integration, which is the minimum time step required for both fluid and solid structures. Usually, the single-resolution method is less computationally efficient in simulations where only fine resolution is required locally. To improve the computational accuracy of local flow fields and reduce the total computational time, Barcarolo et al. [136] proposed an Adaptive Particle Refinement (APR) technique based on the overlapping particle method. Tang et al. [137] applied a similar technique in the MPS framework.

In the APR technique, particles of different resolutions are defined as different sets of particles $L_k$. k denotes the level of particle resolution. Particle splitting occurs when particles of the k th layer enter particle refinement regions. During the splitting process, the particles of the $L_k$ layer are retained, and a new set of particles $L_{k+1}$ is created. After the splitting, the particles in the $L_k$ layer are called mother particles, and the particles in $L_{k+1}$ are called daughter particles. After the mother particles are split, their particle activity is switched off, and they are called inactive particles, which no longer participate in the SPH particle calculation. The inactive particles still follow the flow field and obtain information about the velocity and pressure of the flow field by interpolating from surrounding daughter particles. The mother particles generate active daughter particles that are activated and participate in the SPH particle calculation of flow fields. When the inactive mother particles flow out of particle refinement regions, their activity is activated again, and they normally participate in the SPH particle calculation. At the same time, the refined sub-particles are deleted when they flow out of refinement regions. Information between different particle levels can be exchanged through the Shepard interpolation [33].

Through the above steps, the SPH numerical simulation with multi-level resolution can be achieved (Figure 6), and the calculation accuracy of local flow fields can be improved. Sun et al. [9] and Chen et al. [138] proved that the APR could be easily extended to the simulation of multiphase flows, such as dam-break flows and rising bubbles (Figure 7).

Different from APR, Yang et al. [62] proposed an SPH model with the Adaptive Spatial Resolution technique (SPH-ASR). In the simulation using the SPH-ASR technique, the spatial resolution adaptively varies according to the distance to the different interfaces (Figure 8). Even if the interface becomes complex and evolves rapidly in time, the SPH particle resolution will change adaptively. The SPH-ASR technique for multiphase flows has also been improved by introducing the PST to make particles more uniform. The PST considers variable smoothing lengths and further enhances the SPH-ASR technique by optimizing the adaptive resolution algorithm. The improved SPH-ASR technique has been validated by simulating dam-break flows, droplet formation, and water impact on solid surfaces [139].

Figure 6. Schematic diagram of the APR technique [33,140]. (a) Particle splitting; (b) Particle refinement zone.

Figure 6. Schematic diagram of the APR technique [33,140]. (a) Particle splitting; (b) Particle refinement zone.

Figure 7. SPH simulations of the rising bubble at different time instants with the APR-SPH technique [138]. (a) t = 0.3 s; (b) t = 1.5 s; (c) t = 2.2 s; (d) t = 3.0 s.

Figure 7. SPH simulations of the rising bubble at different time instants with the APR-SPH technique [138]. (a) t = 0.3 s; (b) t = 1.5 s; (c) t = 2.2 s; (d) t = 3.0 s.

Figure 8. Simulations of dam-break flows based on the SPH-ASR technique [62].

Figure 8. Simulations of dam-break flows based on the SPH-ASR technique [62].

3.7. High Performance Computing Techniques

In SPH numerical simulations, precise and complex 3D structures often require fine particle resolution, which leads to high computational costs. With the rapid development of computer technology, the emergence of hardware acceleration technology has greatly improved the computational efficiency of SPH methods and promoted the engineering application of SPH methods. Hardware acceleration techniques can be based on two parallel computing frameworks, CPU or GPU. Mesh methods (FVM) are usually computed using CPU parallelization because they require solving large sparse matrices and implicit time integrals, which are unsuitable or more challenging to compute using GPU parallelization. On the other hand, the weakly compressible SPH method, as a Lagrangian particle method, is based on an ad hoc approach, and the time integration is in an explicit format. These two properties make the SPH method ideally suited for GPU parallelization [34,53,141,142,143]. It has been demonstrated in the literature that GPU parallelization in SPH methods based on the Compute Unified Device Architecture (CUDA) can be ten or even one hundred times more efficient than CPU parallelization at the same scale, which significantly enhances the engineering use of SPH methods [144]. Furthermore, GPU-parallel computing software can run on GPU-equipped PCs and is not entirely dependent on large supercomputers or servers, so it has excellent potential in future engineering applications.

The realization of SPH parallel computing can first be based on the CPU parallel computing framework for programming SPH programs. For example, Marrone et al. [145] built a mixed MPI (Massage Passing Interface)-OpenMP (Open Multi-Processing) programming model. They designed a new 3D parallel SPH solver and achieved the global simulation of a 3D ship, which can provide a reference for constructing a parallel SPH computation model. In addition, simulations can be performed using the current mainstream parallel SPH open-source packages, such as SPHinXsys [146], DualSPHysics [147], and other open-source packages and software (

Table 2. Open-source packages and commercial software based on SPH methods.

Open-Source Packages	Features	Softwares	Features
SPHinXsys	Riemann SPH, Complex boundary, Elastic solid calculation	LS-DYNA	Coupling with FEM
DualSPHysics	GPU Parallelism, High precision, High efficiency,	ABAQUS	Coupling with FEM
AQUAgpusph	GPU Parallelism, OpenCL, Python scripts	nanoFluidX	GPU Parallelism, Complex boundary Mutiphase flows
GPUSPH	GPU Parallelism, High efficiency, Wave conditions	SimArk	GPU Parallelism, Ocean engineering

). They all have good performance in the application of complex 3D engineering problems. Hardware acceleration technology provides a powerful guarantee of computational power to realize large-scale SPH simulations.

4. Multiphase Flows under Atmospheric Pressure

4.1. Influence of Multiphase Effects for Some Typical Problems in Ocean Engineering

Multiphase flows under atmospheric pressure are among the most common multiphase flow problems. In the case of ships sailing at high speed and waves passing into shallows, wave overturning and breaking will occur. At this time, the air above free liquid surfaces will be involved in the water, forming a complex phenomenon of air bubbles interacting with free liquid surfaces [11]. Complex multiphase flow phenomena such as intermixing between oil, water, and gas also exist during the extraction of oceanic oil and gas [148,149]. In addition, small bubbles in the ocean produce jets and splashing droplets when they break up at free liquid surfaces. These droplets can enter the air and be transported over land, promoting the circulation of materials such as salts between the ocean and land [150]. The presence of gas–liquid multiphase flows cannot be ignored in these phenomena of ocean engineering. However, the physical phenomena of bubble movement in liquid are complex, including spiral rising paths, variations of bubble shapes (ellipsoidal, spherical, flattened, etc.), bubble fusion, bubble tearing, and bubble oscillations [12,14].

The single-phase SPH models ignore the presence of gas above free liquid surfaces and only consider the motion of liquids. However, the applicability of these assumptions is limited, and the computational errors can be significant. In the liquid chamber sloshing experiments of Souto-Iglesia et al. [63,151] and Delorme et al. [64], sloshing waves break up and roll over as they move, with large amounts of air being trapped. Subsequently, the trapped air moves together with waves. During the process of waves impacting the walls, the air between water and tank walls has an effect known as an “air cushion”. Many studies have demonstrated that the peak impact load on the walls of a liquid tank differs significantly from the calculated results when a single-phase flow is adopted due to the “air cushion” effect. Measurements of the pressure at liquid chamber walls also show fluctuating impact loads. The compressibility of the trapped air is the dominant factor in this pressure fluctuation. Therefore, many detailed multiphase flow processes must be further studied using numerical simulations.

4.2. SPH Simulations for Multiphase Flows under Atmospheric Pressure

As a Lagrangian particle method, the SPH method does not require tracing the water–gas interface when simulating multiphase flow problems. Moreover, no additional artificial treatment is required to simulate the fragmentation and fusion of multiphase interfaces. However, additional treatment is required when dealing with large density ratios and interface penetration at multiphase flow interfaces. In recent years, the SPH method has been successfully adopted to simulate the multiphase flow. Many researchers have proposed different SPH models and numerical techniques for simulating multiphase flows.

Colagrossi and Landrini [83] studied the SPH governing equations and gave a discrete form of the governing equations suitable for large density ratio multiphase flow problems. Hu and Adams [106] proposed an SPH model for multiphase flows based on a volume approximation. Grenier et al. [114,152] proposed a multiphase interface sharpness model that can effectively prevent penetration at multiphase interfaces. Chen et al. [113] proposed a density filtering technique in the SPH model for multiphase flows. Koh et al. [153] and Luo et al. [153] proposed a new particle method called the Consistent Particle Method (CPM) and applied it to the simulation of multiphase flows. In this method, the spatial derivatives are solved by using a generalized finite difference format. The CPM is consistent with Taylor series expansions and does not use kernel functions, which fundamentally differs from traditional particle methods in the computation of derivative approximations. Of course, the $\delta$-SPH model [59] and the SPH model based on Riemann solvers [99,100,154] also successfully simulate the multiphase flows under atmospheric pressure.

The $\delta$-SPH model and ${\delta }^{+}$-SPH model have been successfully applied to many hydrodynamic problems in the field of ocean engineering, such as gravity wave propagation, dam-break flow (Figure 9 and Figure 10), and liquid tank sloshing [9,84,155]. Lyu et al. [156] and Sun et al. [66] analyzed the problems such as water entries and wave impacts (Figure 11), in which the multiphase effects were analyzed and emphasized. They theoretically analyzed the causes of TI in such problems and combined the ${\delta }^{+}$-SPH model with the TIC technique to overcome this numerical problem.

Based on the work of Parshikov et al. [103], Roubtsova et al. [157] extended the SPH model based on Riemann solvers to simulate violent free surface flows. Recently, Zhang et al. [158] also used this model with a linearised Riemann solver to simulate violent free surface flows. They adopted a simple limiter to deal with the excessive numerical dissipation introduced by Riemann-Solver and obtained good results. Considering the advantages of the Riemann-Solver in dealing with intermittent interfaces, Yang et al. [99] and Hu et al. [159] extended it to a multiphase flow SPH computational model and improved the numerical dissipation term of Riemann-Solver to describe the actual physical viscosity of multiphase flows. The improved Riemann SPH computational model for multiphase flows can accurately capture multiphase interfaces and simulate multiphase flow problems with large density ratios without the addition of artificial techniques. The Riemann-ALE SPH model also has been applied to simulate multiphase flows, water entry impacts, and fluid-structure interaction problems [98].

Grenier et al. [114] proposed a multiphase model containing interface interaction and multiphase flows, which can accurately deal with the discontinuities in physical quantities at interfaces. The model allows the simulation of flows in the presence of both interfaces and free surfaces. The volumetric strain rate equation is used to calculate the evolution of thermodynamic volume, and the density field is evaluated through a Shepard kernel function correction. Zhang et al. [14] adopted the SPH model with summation-based density [107] to deal with multiphase flow problems. An improved prediction-corrected integration method based on this model was proposed, which improves the stability of this model and forms a complete SPH model for multiphase flows. The model is suitable for simulating bubble dynamics problems with large density and viscosity ratios (Figure 12).

5. High-Pressure Multiphase Flows with Strongly-Compressible Effects

High-pressure multiphase flow problems are also common in ocean engineering, where the main focus is underwater explosions [16] and bubble oscillations [160]. In the civil field, high-pressure air guns used for subsea resource exploration also involve the expansion, contraction, and jetting of high-pressure bubbles [161]. The underwater explosion is a strongly non-linear process involving violent expansion with high temperature and pressure. The initial structure and sea state greatly influence the explosion process, making it extremely challenging to study. Numerical simulation is one of the best strategies for studying underwater explosions due to the extremely high cost of experiments. High-pressure bubble oscillation is also a common multiphase flow problem in ocean engineering. High-pressure bubbles are generated at different conditions containing underwater explosions and cavitation inception. The evolution of these bubbles is also a popular topic of current research.

The propagation of shock waves caused by underwater explosions and high-pressure bubble oscillation is characterized by strong compressibility and nonlinearity [162], which requires high accuracy and stability of SPH algorithms. Traditional SPH methods are usually based on incompressible or weakly compressible assumptions, which are difficult to simulate strongly compressible problems such as underwater explosions. In particular, it is difficult to accurately simulate the periodic expansion and contraction processes caused by underwater explosions and bubble oscillations. Therefore, the traditional SPH method needs to be revised and improved. In recent years, some efficient models and techniques have been made in the SPH method for high-pressure multiphase flow problems.

5.1. Transient Strongly-Compressible Problems

In the problems of multiphase flows under atmospheric pressure, where pressure and energy changes are not very violent, the energy equation is often ignored to reduce the complexity of numerical simulations. When the compressibility of fluids does not dominate, and heat does not affect the behavior of fluids, this assumption is valid. However, in high-pressure multiphase flow problems, where small density phases (gas phase) can be strongly compressible, the energy equation cannot be neglected. Furthermore, the EOS based on weakly compressible phases has to be corrected accordingly. In contrast to the traditional weakly compressible SPH method, the pressure has to be linked to the density and internal energy.

In weakly-compressible SPH simulations, the artificial sound speed ($c_0$) is usually used to enlarge the time step and limit the density variation to less than 1%. However, in the strongly compressible condition, the actual sound speed should be used to simulate the compressibility of fluids. In contrast to incompressible and weakly compressible SPH models with a constant smoothing length, in the strongly compressible SPH simulations, at each time step, the smoothing length needs to be updated to match the variable particle spacing due to large volume changes. The use of variable smoothing lengths allows the number of neighboring particles to be adjusted to maintain constant. The particle smoothing length can be adjusted according to the density of particles to improve the computational accuracy and stability of SPH methods. Therefore, there is a need for SPH operators that allow variable smoothing lengths [163]. Oger et al. [164] derived corrected SPH differential operators with first-order completeness, where the integration in derivatives is based on arbitrary kernel functions with non-constant smoothing lengths. Thus, when the smoothing length is variable, the accuracy of SPH methods is not affected. The appearance of SPH models based on Riemann solvers has promoted the application of particle methods to deal with compressible problems. Based on the work of Parshikov et al. [103], Meng et al. [101] proposed the TENO-SPH (Targeted Essentially Non-Oscillatory SPH) model with high numerical accuracy in shock wave capture, which enhances the robustness of SPH models based on Riemann solvers for such problems. Meanwhile, Fang et al. [165] introduced Riemann-Solver into the framework of the axisymmetric SPH model, which is an accurate and stable axisymmetric SPH model. In order to reduce the numerical dissipation caused by the inherent numerical viscosity of Riemann-Solvers, the Primitive Variable Riemann-Solver, which is reconstructed by using the Monotone Upwind-Centered Scheme for Conservation Laws, is embedded into the axisymmetric SPH model. This model allows for the real compressibility of air and a larger steady time step when simulating strongly compressible flows.

In ocean engineering, shock waves often accompany high-pressure multiphase flow problems (Figure 13). The SPH method has made significant progress in accurately simulating shock waves. Wang et al. [166] proposed a reflection-free boundary condition based on the characteristic line principle, which effectively eliminates boundary reflections in underwater explosion shock wave calculations and provides a basis for the SPH method to simulate underwater explosions with high accuracy. Peng et al. [167] used the SPH method to simulate underwater explosions. He investigated the propagation of shock and sparse waves generated by high-pressure bubbles. The solved pressure results are in good agreement with the reference results, which can provide a reference for the feasibility and validity of SPH methods to simulate strongly-compressible transient flows.

Another focus of high-pressure multiphase flow problems in ocean engineering is the evolution of cavitating bubbles. In order to simulate the collapse of cavitating bubbles in water, Pineda et al. [160] proposed a multiphase compressible model within the ALE-SPH framework. The model ensures the continuity of velocity and pressure at interfaces between two fluids and allows for the coupling of interfaces with simple contacts. The model solves the mass, momentum, and energy conservation equations for Eulerian systems by using the EOS. The validity of this model has been verified by the simulation of two-dimensional bubble collapse processes.

5.2. Long-Period Bubble Pulsation Problems

During SPH simulations of strongly compressible flows, the particle volume may change significantly compared to its reference value, leading to significant particle spacing changes. Therefore, variable smoothing length techniques should be used to maintain an appropriate number of neighboring particles. However, in the extreme cases where particles may be over-expanded or over-compressed, the density of gas will expand and compress over a relatively long period, so it is not sufficient to only update the smoothing length and maintain a constant number of neighboring particles. When the SPH method is adopted to calculate high-pressure bubbles, the particles become very sparse due to bubble expansion and very dense due to bubble contraction, both of which can make the results inaccurate. When the volume of bubbles contracts to a minimum, the gas particles are strongly clustered, and the number of particles within the radius of the kernel function increases rapidly. On the other hand, when the volume of gas bubbles expands to its maximum, the distance between gas particles increases, and the number of particles within the radius of the kernel function decreases rapidly.

In the process of bubble expansion, contraction and expansion again, the number of gas particles within the radius of the kernel function changes rapidly. However, due to the weak compressibility of water, the number of water particles within the radius of the kernel function at the edge of bubbles remains almost constant, resulting in a highly uneven particle distribution at water–gas interfaces. It leads to an unsmooth water–gas interface, resulting in reduced accuracy, even making the calculation interrupted. In order to improve the reduced accuracy and stability caused by the violent changes in gas-particle spacing when the SPH method simulates the bubble pulsation, Sun et al. [69,70] established a particle splitting and merging technique based on the SPH Volume Adaptive Scheme (VAS) (Figure 14). The proposed VAS technique has excellent total mass and linear momentum conservation properties. New criteria have been designed based on the ratio of the particle volume and its reference value. These criteria allow the particles to split or merge in the event of over-expansion or over-compression, which guarantees that the particle volume varies within a limited range around the reference value. The performance of the VAS technique has been verified by using a piston test. Moreover, combined with the PST for multiphase flows, particle volumes and distributions remain isotropic and homogeneous in all fluid phases, even in huge volume changes. The method can simulate the pulsation process of an explosive underwater bubble at the boundaries of free liquid surfaces and walls.

Although the SPH method can accurately capture large deformations at water–gas interfaces, it is less efficient in dealing with three-dimensional bubble pulsations because of the large computational domain required to avoid boundary effects. Therefore, Sun et al. [70] proposed an axisymmetric strongly compressible multiphase flow SPH model and implemented this model to simulate the pulsation of underwater explosive bubbles. The processes of expansion, contraction, collapse and jet related to high-pressure bubbles have been accurately captured (Figure 15). The SPH results for the evolution of bubbles and change of the maximum width have been obtained, which are in good agreement with the reference results. It can be observed that the axisymmetric SPH model can accurately predict the dynamic processes of bubble expansion and contraction under different conditions.

Fang et al. [165] combined the VAS technique with the SPH model based on Riemann solvers and applied this method to the underwater explosion problem, which has good computational accuracy. The validity of SPH methods applied to underwater explosion and bubble dynamics has been verified, which provides a new solution for the further study of the underwater explosion mechanism.

6. Multiphase Flows with Phase Change

Multiphase flows in ocean engineering, in addition to the previously mentioned water–gas multiphase flows, also contain phase-change multiphase flows. In ocean engineering, phase changes are mainly found in cavitation and solidification phenomena, which cannot be ignored. Different from the previous water–gas multiphase flows, phase-change multiphase flows involve changes in material properties and have been a challenging topic in SPH simulations. In recent years, the simulation of phase-change multiphase flows has gradually been improved with the proposal of different phase-change models. This section will review the progress of SPH simulations for cavitating multiphase flows and solid–liquid phase-change multiphase flows.

6.1. Cavitating Multiphase Flows

6.1.1. A Brief Introduction of Cavitating Multiphase Flows

Cavitation is an important and complex hydrodynamic phenomenon [170,171,172,173]. A reduction in pressure of a stationary or flowing liquid at a certain temperature can lead to vaporization, which is known as cavitation. Cavitation, with its obvious three-dimensional flow characteristics and violent non-constant properties, is a physical phenomenon that occurs only in liquid, which is widespread in hydraulic equipment, ship propellers, and ocean engineering [174]. Cavitation usually has adverse effects, such as changes in equipment operating characteristics, performance degradation, vibration, noise, etc. The significant destructive effects of cavitation are the pressure shock generated by the collapse of bubbles and the stripping damage to solid walls by micro-jets [174,175]. Solid walls that suffer stripping damage can cause rough surfaces or even perforations with catastrophic consequences [176]. As a result, cavitation is a physical phenomenon that engineers often seek to avoid in ocean engineering or during ship design [177,178,179,180]. The problem of cavitation has been a key topic in the field of hydrodynamics.

The study of cavitation phenomena is mainly determined by experiments using physical models. With the help of experimental results, it is possible to optimize the design of body shapes or to forecast the conditions under which cavitation is likely to occur. However, it is difficult to control the physical parameters of the fluid for each physical experiment due to the effects of gas nuclei, surface tensions, etc. Moreover, the cavitation problem is highly complex, and the influence of various factors on cavitation inception is not yet independent, which makes the results obtained from physical model experiments different from actual situations. Numerical simulations are becoming an important tool in the study of cavitation flows. With the development of computational tools, the numerical simulation of cavitation flows has also made significant progress [181].

6.1.2. SPH Simulations of Cavitation Problems

Although some progress has been made in numerical simulations for cavitation problems, most numerical simulations have been carried out within the framework of Eulerian grid-based methods. However, some advanced CFD techniques, such as Adaptive Mesh Refinement (AMR), could lead to the non-conservation of physical quantities at interfaces, which could influence the original conservation of Eulerian grid methods. SPH methods have unique advantages in simulating multiphase flow problems. However, there seems to be no reliable cavitation model in the SPH framework, and existing SPH methods are often limited to studying unmixed multiphase problems.

With the improvement of SPH methods, such as pressure field improvement, GPU acceleration, PST, and multi-resolution techniques, the SPH method has become a powerful tool for solving real industrial problems. Gradually, researchers have adopted the SPH method to simulate cavitation problems [182]. In 2020, Kalateh et al. [183] proposed a cavitation model within the framework of SPH. In this work, the fluid domain is initially set by liquid particles. When the pressure of each particle is below the saturation vapor pressure of the liquid, it is considered a cavitating particle, and its phase changes from liquid to vapor. Furthermore, when the pressure value of each particle exceeds the saturated vapor pressure, its phase changes from vapor to liquid. This model can simulate the cavitating multiphase flows with a pressure drop related to the cavitation inception. However, this method simply uses a two-component model to simulate cavitation flows and ignores the effects caused by the vapor–liquid mixture that is important for cavitation problems.

Lyu et al. [73] proposed a feasible phase-change model within the WCSPH framework (see Section 3.3). They introduced the VAS technique to deal with the strongly compressible processes in the cavitation process, which increases the numerical accuracy and stability of SPH simulations. The basic Smagorinsky model was chosen as the LES model for turbulence simulations. Three benchmark cases of cavitation flow in a nozzle tube, cavitation flows near a small-angle hydrofoil, and cavitation flows in a large-angle hydrofoil have been tested, proving the satisfactory performance of this SPH model in simulating cavitating multiphase flows (Figure 16).

6.2. Solid–Liquid Phase-Change Multiphase Flows

The solid–liquid phase-change multiphase flows, widely existing in nature and engineering problems, have attracted extensive attention in recent decades. The freezing of river water and the melting of metals both involve solid–liquid phase changes. In ocean engineering, the icing phenomenon is a typical solid–liquid phase-change problem. The process of solid–liquid phase change has always been a challenging research topic in natural and engineering problems such as the freezing of seawater, the melting of glaciers, and the work of icebreakers [184].

6.2.1. Numerical Simulations of Solid–Liquid Phase-Change Problems

Currently, numerical simulations for the seawater icing phenomena mainly adopt the DEM [185,186]. The DEM is suitable for simulating the ice movement and fracture but still has many difficulties in simulating multi-physical processes, especially in the simulation of icing and melting processes, which needs to couple with other numerical methods [187]. Eulerian grid methods have advantages in the simulation of icing and melting processes, but the simulation of icing phenomena includes not only phase-change processes but also the movement of ice in water, so Eulerian grid methods have challenges in the simulation of floating ice.

As a Lagrangian method, the SPH method is inherently advantageous in the simulation of large deformations. By adding a temperature term to the governing equation and using a suitable phase-change model, the SPH method is advantageous in the simulation of solid–liquid phase-change problems [188]. Currently, the SPH method is mainly used to simulate the solid–liquid phase-change phenomena of metal melting problems [189,190]. Duan et al. [191] developed a new scheme to deal with the high-viscous fluid, which can successfully simulate the corium spreading and crust formulation. Weirather et al. [192] proposed a numerical model based on the SPH method in a multiphase and weakly-compressible formulation, which can simulate the melting process of laser beams. Zhang et al. [193] adopted SPH methods to study the melting ceramic droplets impinging and solidifying on inclined solid surfaces.

The metal melting/solidifying problems have similarities with the icing problems. In recent years, researchers have gradually adopted the SPH method to simulate icing phenomena. Monaghan et al. [194] firstly applied the SPH in the solid–liquid phase-change problem. Their work studied the freezing of one-component and two-component systems without fluid dynamics. Farrokhpanah et al. [195] proposed a new and fast SPH formulation to model heat conduction with solidification and melting. Wang and Zhang [196] simulated the phase-change process coupled with the thermal flow based on the SPH. Cui et al. [75] proposed an SPH numerical framework capable of simulating SLD (Supercooled Large Droplets) impingement and solidification at in-flight icing conditions. Wang et al. [197] proposed a highly accurate, multi-resolution particle method to simulate the solid–liquid phase-change process coupled with the thermal flow. Instead of introducing the latent heat in the governing equation, the heat equations for solid and liquid phases can be solved separately.

6.2.2. SPH Simulations of Icing Process

During the simulation of the icing process, if a liquid droplet adheres to the surface of solids, the first step is to build a contact angle model to simulate the pattern of the droplet when it comes into contact with the surfaces of different solid materials [74]. There are two general ways of modeling contact angles in SPH method simulations. In the first way, a model similar to the intermolecular forces is adopted. The contact angle is used as an input parameter in combination with surface tensions, which can be regarded as a way to characterize the rejection at the interface between liquid and solid; In the second way, the contact angle line is geometrically adjusted using a modified normal vector of the contact angle line or by creating a virtual interface [198].

The key to solid–liquid phase-change simulations is the simulation of the phase-change process. In the construction of an ice-water phase-change model, three points generally need to be satisfied [75]: firstly, the model can achieve the simulation of the heat transfer process; secondly, it can predict the liquid–solid evolution process; and thirdly, it can predict the behavior of the captured liquid–solid interface. The third point can be more easily achieved with the SPH method. For the first two points, the introduction of the heat transfer equation and the latent heat model (Section 3.3) are required.

When the viscous force is not considered, the heat transfer equation is introduced to monitor the temperature change process [75].

$$\frac{De}{Dt}=\frac{1}{\rho }\nabla ·(k\nabla T)-\frac{p}{\rho }(\nabla ·\mathit{u})$$

(18)

The discrete form of the heat transfer equation is as follows [75]:

$$\begin{array}{c}\frac{D}{Dt}\left(C_{vi}{T}_i\right)=\frac{1}{{\rho }_i}\sum _j\left(k_i+k_j\right)\left(T_i-T_j\right)·\frac{\left({\mathit{r}}_i-{\mathit{r}}_j\right)·\nabla W_{ij}{V}_j}{{∥{\mathit{r}}_i-{\mathit{r}}_j∥}^2}-\frac{p_i}{{\rho }_i}\sum _j\left({\mathit{u}}_j-{\mathit{u}}_i\right)·\nabla W_{ij}V\hfill \end{array}$$

(19)

where e, $C_v$, k, and T are the energy, specific heat, thermal conductivity, and temperature of particles, respectively. The other physical quantities in the equations are defined similarly as in Equation (1), Equation (4), and Equation (5).

Figure 17 shows the simulations of the droplet icing process using the SPH method. The simulation results based on the SPH method and the latent heat model are in good agreement with the experimental results [74].

7. Granular Multiphase Flows

7.1. A Brief Introduction of Granular Multiphase Flows

Compared to pure fluid and solid, granular multiphase flows are more complex to model because of the complexity of physical processes and the number of factors influencing them. In ocean engineering, the granular multiphase flow usually refers to the water-sand two-phase flow. The water-sand multiphase flows are solid–liquid multiphase flows with full spatial coexistence. The solid particles are spatially present in the fluid, and the water flows are spatially present in solid particles. The research target of water-sand multiphase flow problems is the sediment movement in water flows. The sediment movement is a widespread and important natural phenomenon within rivers and coastal areas. It is a critical dynamic factor in natural processes such as river evolution, channel sedimentation, and coast erosion [199,200]. In practical engineering problems, the multiphase flow of water and sand is a phenomenon that cannot be ignored. The siltation in channels is a common problem in the construction and maintenance of harbors, which is related to various complex factors such as tidal currents, waves, incoming rivers, and sediment sources. It is challenging to solve correctly in practical engineering.

The sediment problem has been given full attention in engineering problems, and the research about sediment movements has gradually formed an important branch (sediment dynamics). The traditional sediment theory classifies sediment moving in water into bed-load sediment and suspended sediment, which is studied using different methods and models [201]. Suspended sediment refers to solid particles which interact spatially with the fluid and are suspended in the fluid. Bed-load sediment refers to the solid particles, such as the sediment on the surfaces of seabeds, which do not interact with the entire space of liquid and are mainly affected by fluid shear force. The forces and motions of sediment are mainly divided into particle–fluid and particle–particle interactions. Particle–fluid interactions are primarily driven by water currents, which include turbulent forces, differential pressure, drag and lift forces on the suspended sediment, and shear forces on the bed-load sediment. Particle–particle interactions, driven primarily by gravity, include inter-particle collisions, inter-particle contact, slip, and friction. Similarly, sediment has a counterforce on water flows, and affects the turbulent state of fluid [135]. The diffusion model is one of the most widely used models in studying sediment movements. At present, commercial software such as FLUENT, MIKE, DELFT, and ADCIRC, which are widely used internationally, all use the diffusion model to simulate the sediment movement [202].

In general, the diffusion model is theoretically simple and computationally convenient, which can meet the requirements of practical engineering problems under certain conditions. However, it is not suitable for studying the mechanism of sediment movements due to the excessive simplification and dependence on empirical formulas. The water–sand movement is a typical two-phase flow problem. The two-phase flow models, which distinguish the two phases and study their interactions, can provide mathematical and mechanical perspectives for studying sediment movements [199,202,203]. The SPH simulations of the water–sand two-phase flows also are mainly based on the two-phase flow models.

7.2. SPH Simulations of the Water–Sand Two-Phase Flows

Many researchers used the SPH method to simulate water–sand two-phase flows [77,204,205], and have achieved good results. Ghaitanellis et al. [77] simulated the dam-break flow on sand beds. As shown in Figure 18, a good agreement is obtained between SPH simulations and experimental snapshots. Sun et al. [76] proposed a DEM-SPH model for water–sand two-phase flows, where the solid and fluid phases are coupled using a local averaging technique that considers lift forces and interphase drag forces. They proposed a variational formulation of pressure-based interphase interactions to ensure conservative properties. At the same time, they proposed a boundary force model that unifies the boundary descriptions in both phases without introducing additional boundary particles, which can reduce computational time. However, the SPH-DEM method requires a smaller computational step between two computational steps, leading to lower computational efficiency, and the information transfer between the two methods needs to be handled precisely. Ulrich et al. [132] extended the application of SPH methods to full-scale ocean engineering problems. They proposed an SPH model, which can mimic the interactions between floating bodies, structure-water–soil interactions, and ship propeller flows.

Bertevas et al. [126] proposed an SPH formulation based on the classical two-phase mixture model, which can simulate the turbulent sediment transport and the sediment disturbances generated by the moving equipment. The governing equations are adopted in a Lagrangian, weakly-compressible SPH framework, and the turbulence is modeled by a RANS approach. The adaptive boundary conditions for shear stress and turbulent quantities are implemented to account for laminar or turbulent flows. The complex rheological behavior of clay water–sediment mixtures can be modeled using a volume fraction and a shear rate-dependent viscosity. Compared with experimental results, the simulation results of the SPH formulation agree well with the existing results.

Zubeldia et al. [133] proposed a two-phase numerical model using the SPH method, which can simulate the scouring of two-phase water–sand flows with large deformation. The study bridges the gap between the non-Newtonian and Newtonian flows by proposing a model that combines yielding, shear, and suspension mechanics. The numerical results of a dry-bed dam-break flow over an erodible bed are in good agreement with experimental data. Pahar et al. [134] proposed a coupled solenoidal ISPH model to simulate the sediment displacement in erodible beds. The coupled model has been validated by three dam-break flow cases with different initial conditions of movable beds, and the simulation results are in good agreement with the experimental data.

Zhu et al. [199] proposed a three-dimensional multilayer SPH model applicable to coupled water–sand dynamics problems based on the mixed theory. He developed two mathematical models based on surface density and intrinsic density, which adopted two overlapping SPH particle layers to represent the two phases independently. The water phase is modeled as a weakly-compressible Newtonian fluid. The sand phase is modeled using an elastoplastic propriety model, which provides more accurate numerical results and more stable simulations.

For the problem of dam-break flows impacting riverbeds, Shi et al. [17] carried out the simulations using a two-phase SPH model with a single set of SPH particles carrying a sediment volume fraction and obtained the simulation results in Figure 19. The simulation results are in good agreement with the experimental results, which proves that the two-phase SPH model proposed by Shi et al. [17] can accurately simulate the process of dam-break flows impacting riverbeds.

8. Summary and Outlook

This paper reviews recent advances in SPH simulations for multiphase flows. With the improvement of SPH models and numerical techniques, the precision and applicability of SPH methods have been improved daily. The SPH method has been widely used in the simulations of multiphase flow problems such as wave breaking with air entrapment, bubble rising and coalescing, underwater explosions, high-pressure bubble oscillations, cavitating flows, icing process, and water-sand mixing flows. It has obvious advantages in accurately simulating the evolution of multiphase flows with large deformations and violent movements.

In future research, there is still space for the further improvement of SPH methods in several areas:

• In terms of computational accuracy and efficiency, the completeness of SPH methods and the order of numerical convergence need to be further improved. The multi-level resolution technique also needs to be improved to achieve computational accuracy with a smaller number of particles.
• In strongly compressible multiphase flow problems, there is still a need for further development of adaptive particle volume and reflection-free boundary techniques under extreme compressible problems, as well as more accurate energy calculation methods. The temperature term needs to be added to the governing equations, which can accurately simulate the shock wave development and energy changes in strongly compressible problems.
• In terms of application areas, turbulence models, cavitation models, and water–sand mixture models within the framework of SPH theory need further development. The simulation ability in high Reynolds number flows, phase-change problems, and solid–liquid mixed media problems need to be further enhanced. SPH models of water–air and water–sand mixtures based on the volume fraction need to be further developed. The simulation of water-bubble mixing flows with high-speed boats and the simulation of sediment movements with the coastal evolution both require further development.
• At present, several multi-resolution techniques are hard to be implemented on GPU, which limits their wide applications in engineering simulations. Therefore, in future studies, these techniques need to be further improved in terms of compatibility with GPU parallelization.