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Violent Wave Impacts and Loadings using the δ-SPH Method

Violent Wave Impacts and Loadings using the 5-SPH Method

Matteo Antuono, [1] [2] Salvatore Marrone and Andrea Colagrossi


Modelling of physical problems with violent impacts and strong fluid-structure interaction often represents a demanding challenge for many numerical schemes. The main difficulties arise from the occurrence of large deformations of the air-water interface (due to wave breaking events) and from the imposition of the boundary conditions along it. Differently from mesh-based schemes (which need specific techniques to model the free-surface evolution), particle methods can directly track the air-water interface thanks to their Lagrangian structure. In fact, numerical particles can be regarded as movable computational nodes in the bulk fluid. A further advantage of such schemes is the fact that it is possible to model solid bodies and boundaries through particles as well. This allows for a straightforward description of moving solid profiles in comparison to many grid-based schemes which generally use immersed boundaries (see, e.g., Peskin,1972; Mittal and Iaccarino, 2005), overset grids (Carrica et al., 2007; Muscari et ah, 2011) or need to implement remeshing algorithms during the evolution (e.g., Lohner et ah, 2008).

A promising particle method which has proved to be accurate and robust in a wide variety of engineering applications is the Smoothed Particle Hydrodynamics (SPH) scheme. This is a meshless Lagrangian scheme whose formulation relies on two main steps: (i) the representation of the continuous differential operators through convolution integrals with a compact-support kernel function, (ii) the discretization of such convolution integrals into a finite set of elementary fluid particles. SPH was initially conceived in astrophysics for the evolution of gaseous clouds (see Gingold and Monaghan, 1977; Lattanzio et ah, 1989) and, later, was applied to hydrodynamics (Monaghan, 1994; Monaghan, 1996; Colagrossi and Landrini, 2003). Because of this historical path, the first SPH models were derived under the assumption that the fluid is weakly-compressible, i.e., that the density variation remains below 1% during the flow evolution. The density field is then used to compute the pressure field through the state equation for barotropic fluids. Through the years, however, several variants of SPH have been defined on the hypothesis that the fluid is incompressible, this being a more natural assumption in hydrodynamics (though not the most convenient for SPH, as we will discuss later). To date, the distinction between weakly-compressible and incompressible models still represents the main classification of the SPH schemes.

A further classification for applications in the hydrodynamic context can be achieved with respect to the modelling of the air phase. The air-water interface can be modelled in SPH either by a single-phase approximation (i.e., only the water phase is discretized) or by a two-phase approach (both air and water domains are discretized). The former allows to reduce the CPU costs of the simulation whereas the latter is needed whenever air-cushioning effects are of interest in the simulation.

In the present chapter we will only deal with single-phase SPH schemes; in particular, we will focus on weakly-compressible variants. One of the main advantages in this case is that the dynamic boundary condition along the free surface is implicitly satisfied during the flow evolution (see Colagross et ah, 2009; Colagrossi et ah, 2011). This implies that it is not necessary to detect the free surface, leading to a considerable reduction in computational cost. A further fundamental difference between compressible and incompressible schemes is that the former are explicit in time while the latter are implicit and rely on the solution of a Poisson equation for the pressure field. The linear system that stems from the above equation requires a large computational cost which essentially cancels out the advantage for a larger time step during the numerical simulations in comparison to the weakly-compressible variants. In turn, being explicit in time, the latter schemes are more easily parallelized and show good scalability properties (see Crespo et ah, 2011; Oger et ah, 2016).

The main drawback of the weakly-compressible models is represented by the generation of a spurious numerical acoustic noise that affects both the pressure and density fields. In the SPH literature, several attempts have been made to overcome this issue. In Colagrossi et ah (2003) a filtering of the density field was suggested through the use of a Moving Least Squares (MLS) integral interpolation. This approach gave good results for confined flows but, in the presence of a free surface and for long-time simulations, it did not conserve the total volume of the fluid system, since the hydrostatic component of the pressure field was filtered improperly (see example, Sibilla, 2007). An alternative approach was proposed by Vila, 1999 and Moussa and Vila, 2000, who studied the convergence of SPH schemes using approximate Riemann solvers to model the particle interactions. From their works, a family of SPH schemes based on Riemann solvers was developed (see Marongiu et ah, 2010; Koukouvinis et ah, 2013; Oger et ah, 2016). These models generally provide accurate results but, unfortunately, do not allow an exact quantification of the dissipation caused by the numerical scheme that is generally quite large.

Exploiting the theory of Riemann solvers, Ferrari et ah, 2009 defined a numerical diffusive term based on the use of a Rusanov flux. This term was added in the continuity equation to reduce the numerical noise affecting the density field. A similar approach was followed in Molteni and Colagrossi, 2009 where the diffusive term was modelled as the Laplacian of the density field. It is worth noting that a similar term was obtained in Clausen, 2013 by rewriting the continuity equation in terms of the pressure field. In particular, this was achieved by imposing the entropy variation to minimize the density fluctuations, this corresponding to an approximation of an incompressible flow. Unfortunately both the Ferrari et ah, 2009 and Molteni and Colagrossi, 2009 schemes proved incompatible with the hydrostatic solution in the presence of a free surface, since the adopted diffusive terms introduced non-zero contributions close to the interface. To circumvent this issue and, at the same time, maintain the positive features of the above-mentioned diffusive schemes, Antuono et ah, 2010 proposed a correction to the diffusive term of Molteni and Colagrossi, 2009. The corrected term proved to be compatible with the hydrostatic solution and to properly smooth out the numerical spurious oscillations from the pressure and density fields.

The scheme described in Antuono et ah, 2010, which is now known as the d-SPH scheme, has been widely inspected from both a theoretical (see, for example, Antuono et ah, 2012; Antuono et ah, 2015) and a numerical point of view (Marrone et ah, 2011; Antuono et ah,

2015; Bouscasse et al., 2013; Bouscasse et al., 2013), proving to be an accurate, robust and reliable model for a broad variety of applications in fluid dynamics (see, e.g., Canelas et al., 2016; Meringolo et al., 2015; Crespo et al., 2017; Altomare et al., 2017). For these reasons, in the present chapter we will exclusively deal with the <5-SPH scheme, providing details on the implementation of the model, the procedure to assign boundary conditions along solid profiles and, finally, showing some relevant applications.

  • [1] CNR-INM. Institute of Marine engineering. Via di Yallerano 139. Rome, 00128, Italy.
  • [2] Corresponding author: This email address is being protected from spam bots, you need Javascript enabled to view it
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