Modelling solid bodies
The presence of solid profiles causes a cut of the kernel domains of particles that are close to the body. To avoid inconsistencies and to assign the correct boundary conditions, the “missing” volume in the incomplete kernel domain has to be properly modelled. Incidentally, we underline that such a cut also occurs for particles close to the free surface but, in that case, this does not induce any problem, since the weakly-compressible SPH implicitly satisfies the dynamic boundary condition along the interface.
In the SPH literature several techniques have been proposed to impose the correct boundary conditions along solid profiles. Early techniques relied on repulsive (Monaghan, 1989) or dynamic particles (Dalrymple and Knio, 2001). Later on ghost fluid approaches (Randles and Libersky, 1996; Colagrossi and Landrini, 2003) were proposed and adapted for general body shapes as in the case of the fixed ghost particles Marrone et ah, 2011 or dummy particles Adami et ah, 2012. More recently semi-analytical (Kulasegaramet ah, 2004; Leroy et ah, 2014) approaches and methods based on the evaluation of the flux normal to the wall (Marongiu et ah, 2012; De Leffe et ah, 2009) have been proposed. Due to its flexibility, robustness and accuracy, the fixed ghost particle method is one of the best suited technique for the applications considered in the present Chapter. For this reason, we just restrict the discussion to such a method and address the interested readers to the above mentioned works for alternative approaches.
Before proceeding, we highlight that free-slip boundary conditions are imposed along the solid profiles. In fact, for the applications described here, the presence of the boundary layer may be neglected. This also means that the viscous term introduced in system (12) only represents an artificial bulk viscosity.
The ghost-fluid method
In the present section we describe how to enforce the appropriate boundary conditions on the body surface by using the ghost-fluid technique. This basically comprises two steps: (г) the solid domain is modelled through a set of “imaginary particles” (hereinafter denoted as “ghost particles” and labelled with the subscript “s”), (гг) the fluid fields (that is, velocity, pressure) are extended on such fictitious particles through proper mirroring techniques. In particular, different mirroring techniques are adopted to enforce different boundary conditions (e.g., Dirichlet or Neumann conditions).
At step (г), the solid surface is discretized in equispaced body nodes and several layers of ghost particles are disposed inside the solid up to a radius of the kernel domain (see the sketch in the left plot of Figure 2). At step (гг), the velocity and pressure assigned to the fixed ghost particles, namely (us,ps), are computed by using the values interpolated at specific nodes internal to the fluid and uniquely associated with the fixed ghost particles. Hereinafter, the interpolated values are indicated through (u*,p*). More details about the way to dispose the ghost particles and define the interpolation nodes can be found in (Marrone et al., 2011).
The pressure field ps is mirrored on the fixed ghost particles to enforce the following Neumann condition (derived from the momentum equation):
where / is a generic body force and щ, is the velocity of the solid boundary (for details see Marrone et al., 2011). This leads to:
where r* indicates the position of the interpolation node.
Conversely, the velocity field is the object of a specific treatment. As sketched in the right plot of Figure 2, the ghost velocity us depends on both u* and U/,, the latter being the velocity of the nearest body node. De Leffe et al., 2017 found that different mirroring techniques have to be used to evaluate different differential operators (for example, the mirroring technique for the velocity divergence is different from that used for the Laplacian of the velocity). This is needed to avoid inconsistencies, loss of accuracy or, in the worst case, instabilities. The specific mirroring techniques depend on the components of u’ in the normal and tangential direction to the solid surface (right plot of Figure 2). In particular, De Leffe et al., 2011 proved that the velocity-divergence operator is convergent and consistent
Figure 2: Sketch of the ghost-fluid approach. Left: Discretization of the ghost-fluid through ghost particles. Right: Mirroring of the velocity.
Figure 3: Sketch of the activated ghost particles at the intersection between the free surface and the body surface.
if the normal component of u* is mirrored in the frame of reference of the solid profile (see Colagrossi and Landrini, 2003), leaving the tangential component unaltered:
Regarding the viscous operator in the system (12), this approximates the Laplacian of the velocity field and, therefore, a different mirroring technique has to be implemented. Since this is actually an artificial viscous term (which is just used to regularize the velocity field), the adopted mirroring technique is designed to give a negligible contribution along the solid profiles, thus resembling free-slip conditions. In particular, we use:
which simply represents a uniform extension of the velocity field in the solid domain.
Before proceeding to the evaluation of forces and torques through the ghost-fluid technique, we highlight that some care has to be given to the numerical treatment of the intersection between the free surface and the body. In fact, in this case some interpolations nodes may be outside of the fluid region and, therefore, may cause an inaccurate mirroring. To avoid this issue, these nodes (and the related ghost particles) are switched off, as shown in Figure 3. A reliable technique for the detection of these nodes is described, for example, in (Bouscasse et al., 2013).