When using a Riemann-SPH model the velocity and pressure conditions at the interfaces can be handled in a more direct way, see, e.g. [106, 143]. For example, the pressure jump linked to surface tension can be explicitly prescribed, as in [142].

Besides, when using the different stabilization techniques (see section 11.2) separated summations for each phase need to be performed in order not to introduce spurious numerical effects at the interfaces, see, e.g. [106].

### 13.1. The choice of the speeds of sound

The choice of speeds of sound in the EOS of the different phases (see section 2.2) is crucial, for CPU cost and stability reasons.

For the liquids, with the weakly-compressible approach discussed in section 12 it is possible to select a speed of sound $c_{0x}$ which is much lower than the real one, but, at the same time, it needs to be sufficiently high to guarantee a negligible effect of the artificial compressibility.

For the gaseous phases this choice is more critical. For example, to simulate a liquid impact problem with air entrapment the real compressibility of the gas needs to be taken into account; in other words the real speed of sound of the gas phase $c_{0y}$ has to be chosen. This leads to a paradox stressed in [54] where the speed of sound of the gas phase can turn out to be much higher than that of the liquid one, i.e. $c_{0y} > c_{0x}$. As remarked in [172], the CPU cost of this kind of simulation would then be dominated by the small time steps required by the speed of sound of the gas.

If, for the problem at hand, the compressibility of the gas phase is not a crucial aspect, then $c_{0y}$ can also be chosen lower than the real one, as for the liquids. In particular, in order not to increase the CPU cost one can then select $c_{0y} = c_{0x}$. However, as shown in [106] for $c_{0y}/c_{0x} \le 1$ and large density ratios $\rho_{0y}/\rho_{0x} \gg 1$, for most of the SPH schemes the CFL stability constant $K_c$ then largely decreases, making the computational costs prohibitive in many cases. If the SPH scheme is stabilized through a Riemann solver this drop of $K_c$ can be avoided. On the other hand, for ratios $c_{0y}/c_{0x} \ge 1$ the opposite behavior is observed where $K_c$ can be increased to higher values if the scheme is not stabilized through Riemann solvers. There is a lack in the SPH literature about these stability problems related to multiphase flows which deserve future analysis.

From the point of view of the computational costs, if the compressibility of the gas phases is not crucial, it is shown in [252] that it can be beneficial to treat both liquid and gas phases as truly incompressible using approximated SPH Poisson solvers.

### 13.2. Simulation of highly-compressible gas phases

In some applications the gas compressibility can be very high and, beside the constraint on the speed of sound reported above, the particle volumes can change so much that it leads to much denser or, respectively, much less dense particle distributions. In principle, this in turn should require variations of

the kernel smoothing length to avoid the number of neighbor particles becoming very high when a gas pocket is highly compressed or, respectively, very low when that pocket becomes very large. In section 3.1 it was underlined that SPH models with variable smoothing length imply an increase in the complexity of the model and in the coding of the solver. These drawbacks can be limited through the use of splitting/merging algorithms which can adapt the number of particles inside a gas pocket when it is submitted to considerable changes of its volume. That strategy was successfully adopted in [237, 238]. In [237] a Volume Adaptive Scheme (VAS) is described which guarantees a limited variation of the smoothing length as well as a distribution of particle volumes $V_i$ where the ratio between $V_i$ and its initial value $V_{i0}$ is maintained within the bounds: 0.4–1.6. In the second work [238] this highly-compressible multiphase flow SPH model is rewritten in cylindrical coordinates in order to simulate axial symmetric bubble dynamics problems.

An example of a result obtained using this VAS is given in figure 14 where the expansion and collapse of an underwater-explosion bubble in the proximity of the free surface is simulated by the multiphase SPH model and compared with experimental pictures, see [238]. For this problem the barotropic hypothesis is still valid, while in the related work [237] when dealing with interactions between bubbles and strong shock waves a more general EOS $p = p(\rho, e)$ was adopted. As stated in section 2.2 the latter kind of flow is not within the scope of the present review.

Note that, although phase change is not a topic of this review, the VAS algorithm was also used in a cavitating flow situation with highly-compressible gas phase in [161].

## 14. Examples of free-surface and multiphase flows solved by SPH models

In this last section some examples of complex free-surface and multiphase flows solved through SPH models are presented. The selection was made by considering some of the classes of flow where SPH has proven to be a competitive method with respect to other numerical methods.

The first example concerns the simulation of sloshing flows. These flows occur when liquid motions are induced inside tanks. This phenomenon is of interest for several branches of engineering. Especially when strong non-linear sloshing dynamics occurs, SPH models have proven to be a powerful simulation tool.

For example figure 15, taken from [31], refers to a comparison between SPH simulation and experimental data for a shallow-water sloshing flow. The filling height $H$ is 3 cm and the length of the tank $L$ is 1 m. The tank is forced to move horizontally with a harmonic law of amplitude $A$ equal to 7 cm and a period of 3 s. With this choice of parameters a breaking bore forms. The bottom-left plot depicts the time history of the wave elevation measured by a capacity-gauge positioned 5 cm from the tank side. In the right plot some experimental pictures are compared to the free-surface particle positions detected during the SPH simulation. This result highlights the ability of
<figure><img src="image_18.png" /><figcaption>**Figure 14.** Expansion and collapse of an underwater-explosion bubble near a free surface. Comparison between experiments (top) and SPH simulations (bottom). Reprinted from [238], Copyright (2021), with permission from Elsevier.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure 15.** Shallow water sloshing flow. SPH solution compared with experiments. Used with permission of De Gruyter, from [31]; permission conveyed through Copyright Clearance Center, Inc.</figcaption></figure>

the SPH models to accurately reproduce this highly non-linear phenomenon. In this work [31] a wide shallow-water sloshing flow panorama is given with several comparisons between SPH and experimental data.

A second topic where SPH models have been successfully applied concerns water-entry problems. This kind of flow is characterized by a large deformation of the air-water interface which takes place in a short time range. During the solid water entry the pressure on its surface is generally characterized by intense localized values quickly moving along the body surface. An example is given in figure 16, taken from [206], where the ditching of a helicopter model is studied both experimentally and numerically. In the figure a fair comparison between SPH and experimental data on the pitching motion during the water entry is reported. The bottom-right plot shows the pressure field and the different layers adopted for the particle refinement in order to resolve the sharp high pressure zone moving along the bottom of the helicopter model.

Another example of a complex water impact problem is shown in figure 17, taken from [73], depicting the interaction of a large wave with an oil rig. The latter is characterized by a complex geometry and the simulation required more than 10⁹

particles which remains one of the largest SPH free-surface flow simulations. It was carried out using 64 Tesla 2090 GPUs. To simulate 12 s of physical time, a total number of 237 342 time steps were performed in approximately 92 h of physical time.

For strong water-solid interactions the mutual coupling between the fluid and the structure can play a relevant role. This is the case, for example, of the tire aquaplaning studied in [111]. In this work the fluid evolution is simulated with the SPH method in order to handle the 3D complex interfaces of the hydroplaning phenomenon, while in the solid domain a Finite-Element method is used for dealing with the deformation of the tire and treating the tire-ground contact, see figure 18.

All the four studies described above were performed with single-phase SPH models, while the four investigations presented below are based on multiphase SPH models. The first example concerns the prediction of the dynamic behavior of a damaged ship in a case of floodwater. To this purpose an experiment was performed with forced roll tests of damaged ship compartments which was then simulated with SPH and MPS multiphase models. Figure 19 shows the comparison
<figure><img src="image_20.png" /><figcaption>**Figure 16.** Ditching of a helicopter model. Comparison between experimental data and SPH simulations. Reproduced from [206], with permission from Springer Nature.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 17.** SPH simulation of a large wave interacting with an oil rig. Reprinted from [73], Copyright (2013), with permission from Elsevier. The full video of the simulation is available at Link Video N1.</figcaption></figure>

between the experiments and the SPH simulations. It is worth noting that during the flooding process large air bubbles are trapped inside and released from the ship compartments, a phenomenon which cannot be considered using a single-phase model.

A second example regards the atomization of liquid films where the breakup mechanism leads to a wide range of droplet sizes and trajectories, making the SPH method suitable to

simulate this kind of complex phenomenon which is still not fully understood in detail. In [33] a direct numerical simulation (DNS) of an experimentally-investigated planar prefilm-ing air-blast atomizer flow is presented. A reliable numerical prediction of such an air-assisted atomization requires considerable computational power. In the SPH simulation the spatial resolution of 5 µm required the use of 1.2 billion particles at the end of the simulation, see figure 20.
<figure><img src="image_22.png" /><figcaption>**Figure 18.** Study of a tire aquaplaning situation. Left: experimental tests. Middle and right: SPH simulations where the particles are colored with the intensity of the velocity field, using an idealized smooth (middle) or real rough (right) ground. Reprinted from [111], Copyright (2019), with permission from Elsevier. The full video of the simulation is available at Link Video N2.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 19.** Forced roll test of a damaged ship compartment. Top: pictures from the experiment at four different time instants. Bottom: multiphase SPH simulation at the corresponding times. Reprinted from [108], Copyright (2022), with permission from Elsevier.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Figure 20.** SPH simulation of the breakup mechanism in a planar prefilming air-blast atomizer. Reprinted from [33], Copyright (2019), with permission from Elsevier. The full video of the simulation is available at Link Video N3.</figcaption></figure>

The detail of bubble dynamics has also been investigated with the method, as in [270] where an oblique bubble coalescence was studied with comparison between an SPH simulation and experiments, see figure 21. In this kind of phenomenon the air-water interface is subjected to fast deformation with reconnection mechanisms which are well reproduced by the multiphase SPH model.

A further example where SPH was adopted to simulate complex bubbly cloud dynamics is reported in [55] where

the authors simulated the fall of a stream of water from a sluice into a pool. As shown in figure 22 through a multiphase SPH simulation, this flow is characterized by the development of bubble clouds. During their rise toward the pool surface the bubble coalescence mechanism leads to a large variety of bubble sizes. The bubble motions induce an intense turbulent regime in the water. The above flow features were described in a famous drawing by da Vinci (RCIN 912660 The Windsor Collection) which inspired this study.
<figure><img src="image_25.png" /><figcaption>**Figure 21.** Simulation of an oblique bubble coalescence. Comparison between an SPH simulation and pictures from an experiment. The color of the SPH particles that belong to the air phase represents the magnitude of surface tension. Reprinted from [270], Copyright (2022), with permission from Elsevier.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 22.** Fall of a stream of water from a sluice into a pool. Left: bottom part of the sheet RCIN 912660 by Leonardo da Vinci, Royal Collection Trust Copyright His Majesty King Charles III 2023. Right: multiphase SPH simulation. View from the bottom of the pool showing the bubbly clouds rising upward toward the pool surface. Reproduced from [57]. CC BY 4.0. The full video of the simulation is available at Link Video N4.</figcaption></figure>

# 15. Conclusion and perspectives

## 15.1. Conclusion

The present review article is dedicated to the last 25 years of research on the smoothed particle hydrodynamics (SPH) method applied to free-surface and multiphase flows. These flows are today the topic of the majority of the studies on the SPH method, which led to its growing success and diffusion in many engineering domains. This is linked to specific features of the method which are examined in detail in the present review article.

Applications of the method mainly target geometrically-complex flows, whose complexity arises either from the geometrical configuration of the fluid interfaces (with fragmentations, reconnections, etc), or from that of the solid boundaries (with contact, small gaps, complex surface details, etc), or from both. In this context, the Lagrangian and meshless features of this particle method make the SPH method a suitable simulation method. In particular, this method intrinsically describes any fluid interface as perfectly sharp and has

no limitation in terms of geometrical domain discretization or evolution.

In the present work the specific issues of the application of the method to these free-surface and multiphase flows are covered through a thorough review of dedicated key articles. These specific issues include especially the modeling of the contact discontinuity at the interface, the use or not of a single-phase approximation, the representation of liquid impacts within a weakly-compressible or an incompressible approximation, the accurate representation of the Lagrangian kinematics of the interface, or the consistent enforcement of interface conditions including energy exchanges.

Besides, the present review article clarifies numerous theoretical and numerical confusions found in the literature, which concern both the physical and numerical particularities of the free-surface or multiphase flows studied, and the core of the SPH method itself. The combination of the Lagrangian and meshless nature of the SPH method, indeed, makes its numerical development and analysis very difficult. In particular, several complementary points of view and lines of development are reported and discussed through the study of the
most meaningful articles in the literature. These complement-
ary point of views concern, in particular, the derivation of the
SPH equations as a Lagrangian mechanical system or through
the discretization of classical PDEs, the hypothesis made for
the liquid phases, either with a weakly-compressible assump-
tion or with an incompressible one, and the choice of using
either a purely-Lagrangian or an ALE framework to advect
the particles.

Different SPH schemes for simulating free-surface and multiphase flows are presented and discussed in the present work, underlining the context in which these models were developed. The first scheme was developed in the framework of Lagrangian thermo-mechanics in which the original SPH method was initially derived to simulate astrophysical problems, starting in the 1970s. There the fluid is considered as a macroscopic system of *N* fluid particles for which a Lagrangian function can be defined, and the dynamics of this particle system is derived directly through classical Lagrangian equations. The latter ensure the conservation of mass, momenta and energy of the particle system, which was one of the main characteristics of the SPH method leading to its success and diffusion in the computational fluid dynamics (CFD) community. Within this context the key role of the definition of the density field through a continuous kernel function is stressed in the present work. For this definition different choices are possible and, consequently, different SPH schemes can be derived. However, it is argued that only one of those schemes in the literature can be used for simulating both free-surface and multiphase flows.

More classically, a second way to derive an SPH scheme
able to deal with free-surface and multiphase flows is through
direct discretization of the NSEs, with the use of integral inter-
polations. This second path is discussed both by considering
the discretization of local equations and through equations
written in conservative form. These discretizations of the gov-
erning equations lead to SPH schemes which present some
small differences that are discussed and contextualized.

Beside the discussion of the derivation of the different SPH
schemes in the last decades, an extensive examination is also
conducted to present how viscous, thermal and surface tension
models were adapted to the SPH context.

In addition, as for classical mesh-based numerical schemes,
SPH schemes also require corrections to eliminate instabilit-
ies, increase robustness and improve accuracy. The different
techniques which are found in the literature in this respect are
presented and discussed.

Regarding liquid phase modeling, SPH models generally
treat liquids as weakly-compressible; it is only quite recently
that efficient incompressible SPH models where derived. A
discussion of the advantages and drawbacks of these two dif-
ferent approaches is provided.

Finally, some examples of free-surface and multiphase
flows solved by SPH models are shown. These selected
examples from the literature consider types of flows which are
difficult to solve with mesh-based solvers, enabling a discus-
sion of the potential advantages of the SPH method.

Summarizing, in the present review article the significant
literature from the last 25 years is coherently linked providing
a global and detailed understanding of the SPH method when
applied to free-surface and multiphase flows. This should
potentially aid meaningful future developments in this specific
context.

15.2. Perspectives

The method is currently effective for simulating
geometrically-complex interface and free-surface flows.
However, complex interfacial phenomena involving, for
example, phase change, are still very rarely covered by cur-
rent SPH models. Furthermore, since the cost of the method
remains generally high for a given level of accuracy, it is
not the best choice in flow regions that are not close to these
geometrically-complex interfaces. It is therefore likely that
three main lines of development will be pursued in the com-
ing years:

(i) The development of robust and accurate coupling algorithms between the SPH method for the flow regions incorporating complex interfaces and mesh-based methods for the rest of the flow.

(ii) The development of adaptive particle refinement (APR) procedures, in general, and specifically close to interfaces.

(iii) Improvements in the method's efficiency, in general, and especially to describe interfaces.

All these lines of development are related to the reduction
of the CPU costs of the SPH method, which are generally
higher than those of mesh-based methods. Therefore, coup-
ling the SPH method with the latter can be a powerful solution,
limiting the domain of the SPH method to fluid regions where
fluid interfaces are subject to large deformations, including
fragmentations and reconnections. In this way, the advantages
of the SPH method to describe interfaces can be maintained
while avoiding its use in other fluid regions, especially far
from interfaces, where mesh-based methods are more suitable,
being more efficient and accurate there. In recent years, sev-
eral successful attempts have been published along this line,
see, e.g. [40, 44, 69, 172, 199, 256].

In addition to limiting the regions of the flow where the
SPH method is used, it is necessary to increase its efficiency
in the vicinity of interfaces or free surfaces. The accuracy of
the computation of the interface normal and curvature is par-
ticularly crucial for an efficient description of surface tension
and for more complex interfacial physics in general, see, e.g.
[166]. This can be achieved for example by corrections that can
increase the order of convergence and/or minimize the error
level of the method. This is especially important when opera-
tions are performed per phase at the boundary, i.e. when only
particles from one side of the interface are used to numerically
evaluate a given quantity of interest.
Indeed, another goal to improve efficiency is to reduce the number of particle interactions, i.e. to reduce the number of neighbors used for the evaluation of differential operators. Examples along this line of development can be found in, e.g. [9, 19, 162, 179, 201, 257]. However, as emphasized throughout this review work, any corrected scheme must be specifically adapted near interfaces, in particular to preserve quantity jumps across the interface, in order to preserve its sharpness naturally provided by the Lagrangian feature of the method. It will be especially the case if one wants to use incompressible schemes whose current development may lead to significant efficiency gains by using less restrictive time step constraints. Besides, efforts made to take advantage of modern HPC hardware architectures, including GPUs, must also be mentioned. Large scaling factors can be achieved for simulations involving particle numbers up to billions, as emphasized in the previous section.

Another crucial aspect for the development of the applicability of the method towards free-surface and multiphase flows is the ability to refine the spatial discretization, i.e. the particle distribution, in the vicinity of interfaces. Indeed, problems involving many complex interfaces, such as atomization or very fragmented industrial flows like, e.g. the dynamics of oil in a gearbox, require a high level of adaptivity of the local discretization to reach an acceptable level of efficiency. Although great progress has been made with APR techniques, their versatility has not yet reached that of AMR algorithms in mesh-based methods that have allowed a significant improvement of fluid dynamics simulations in recent years, especially in the context of free-surface and multiphase flows, see, e.g. [7, 78, 146, 210, 243].

This APR together with the method's efficiency improvement are key factors to handle simulations with phase changes, such as evaporation of liquid droplets impacting on heated surfaces, see, e.g. [145]. As for phase changes, several SPH articles can be found in the literature that address changes between liquid and solid states, see, e.g. [5, 74, 94]. For phase changes between liquid and gas, on the contrary, few articles exist. The additional difficulty for the method in that case lies in the large density change that occurs at the interface, in factors of the order of 1000–10 000. In fact, if one wants to preserve the key advantage of the method of maintaining sharp interfaces in this context, i.e. to avoid techniques using mixed quantities like the VoF method, one needs to be able to cope with a very large volume change of particles that are evaporated, respectively liquefied. Advanced algorithms dealing with dynamic particle redistributions, involving particle merging, splitting, etc are therefore needed. Such algorithms exist, but are not yet widely applied in that context of large volume change. Some attempts in this direction can be found in [127, 161]. Besides, the sharp physical jumps at the interface in case of phase change can also be incorporated into Riemann solvers, as done with some mesh-based methods, see, e.g. [126].

The future of the SPH method for free-surface and multiphase flows therefore probably lies in the ability of researchers and developers to concentrate the numerical cost and efficiency of the method in the immediate vicinity of the

interfaces, adaptively and possibly in combination with mesh-based methods in the rest of the fluid domains.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

We warmly thank our close collaborators for their careful review of this work, in order of duration of their collaboration with us: Guillaume Oger, Matteo Antuono, Salvatore Marrone, PengNan Sun, Julien Michel and Chiara Pilloton.

## ORCID iDs

David Le Touzé https://orcid.org/0000-0003-3315-7306

Andrea Colagrossi https://orcid.org/0000-0001-7123-3272

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