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Evaluation of Forces and Torques through the ghost-fluid method

The forces on solid bodies are generally calculated by using two main approaches: (г) the integration of the local stresses along the body surface (see, e.g., Fourev et ah, 2017), (гг) the definition of a volume integral over the ghost particle domain Doling, 2005; Bouscasse et ah, 2013. Consistent with the introduction of fixed ghost particles in the previous section, we only deal with the latter approach (see Bouscasse et ah, 2013 for more details).

To find out the formulation for the global loads exerted by the fluid on solid structures, it is convenient to develop the analysis at the continuum level and, later, derive the discrete equations. In the following, the fluid and solid domains are denoted by il f and Qs respectively while (T) indicates the smoothed stress tensor. Then, the global force on the body is expressed by:

where n is the unit outward normal to the solid profile. Assuming the flow field to be mirrored on the solid body through a suitable ghost-fluid technique, the stress tensor can be decomposed in:

where the starred variables indicate quantities mirrored over the solid domain Os. Substituting (21) into (20) and using the divergence theorem and the symmetry properties of the kernel function, we obtain the following equality:

where V denotes the differentiation with respect to the position r. The terms of order 0{h) indicate the contributions due to the presence of the free surface. These terms are small since, by definition, the tension along the free surface is zero. When Equation (22) is discretized, it reads:

where г and j denote quantities associated with the fluid particles and the ghost particles respectively. One of the advantages of Equation (23) is that it directly uses the flow quantities mirrored inside the body while it does not require interpolation on body nodes (i.e., along the body profile). For this reason, it is simpler and faster to use in practical applications. Since the inner summation of (23) approximates the divergence of the stress tensor, in practical simulations it is sufficient to substitute the corresponding operator of the SPH scheme at hand. In the present case, this leads to:

Note that, despite the fact that fluid is assumed to be inviscid, the artificial viscosity term has to be included in the formula above. This is required to ensure the correct energy balance between the solid and fluid phases (see the next section for more details). In any case, the mirroring technique defined in Equation (19) guarantees that the viscous contribution in formula (24) is generally small.

The evaluation of the torque TflUjd-solid acting on the solid body is derived following the same approach shown above. Let us consider a fixed point Го■ Then, the torque with respect to it is:

and, following the same procedure adopted for the evaluation of the force, it is possible to rearrange the above expression as follows:

Bv analogy with formula (24), the above equation is discretized as follows:

The above expressions for the force and torque are reminiscent of the technique proposed in (Monaghan et ah, 2003) and (Kajtar and Monaghan, 2008). In any case, we underline that, apart from the different enforcement of solid boundary conditions, in those cases the formulation is directly obtained from a momentum balance between the fluid and the repulsive body particles. Conversely, in the present case, the global loads are derived from the evaluation of the stress tensor on the body surface by means of a proper ghost fluid extension in the solid region.

Algorithm for fluid-body coupling

In the present section, we briefly describe an algorithm for the coupling of the fluid-body dynamics. The solid dynamics are modelled by Newton’s law of motion which, for the sake of simplicity, is introduced here in a 2D framework:

where Va and Hq are the velocity of the centre of gravity and the angular velocity of the body, M and Ig are the mass and the moment of inertia around the centre of gravity and, finally, Ffluid-solid is the hydrodynamic force acting on the body. Here, Tfluid-solid is the projection of the hydrodynamic torque along the unit vector к normal to the 2D plane, that

is Tfluid— solid = ^fluid-solid • fc.

The dynamical state of the system made of fluid particles, ghost particles and body nodes can be expressed through the vectors yf, ys and уь respectively:

while the dynamical state of the rigid body is expressed by:

where rc and вс are respectively the position vector of the centre of gravity and the related angle of rotation. The coupling between the two systems from Equations (12) and (28) is given by:

where the last two equations represent, respectively, the dependence of the ghost state [see Equations (17), (18) and (19)] and of the body nodes state from the rigid motion equations.

Since the d-SPH equations march in time using an explicit scheme (a fourth-order Runge- Kutta scheme in the present case), the same integration scheme is adopted for the whole system (31). Note that the acceleration of the body appears on both sides of the system (31). This is common in explicit schemes for fluid-body coupling. In particular, in potential flow solvers the body acceleration is generally taken into account through the added mass term (see, e.g., Vinje and Brevig, 1981), enabling one to move such an acceleration term from the right side to the left side. The added mass approach is also applied in other numerical solvers as an under-relaxation correction in the body motion (see, e.g., Adami et al., 2012). In the present scheme the acceleration of the body on the right side is taken from the previous Runge-Kutta substep. This procedure is justified by the use of a very small time-step that is required by the weakly-compressible assumption, as briefly recalled below.

At the generic time instant tn, the state vector уь is determined through yg (predicted at the previous Runge-Kutta substep) and yg. Then, the ghost fluid state ys is obtained through the interpolation on the fluid particles, у/, and the mirroring procedure (which requires yi,, as discussed in Section 4.1). The global loads (F,T)fuid-solid are evaluated through Equations (23) and (27) and yg is obtained by (28). Finally, the interaction between fluid and ghost particles, namely y'f, is computed through the equations of system (12). The iteration substep ends with the integration of yg and y'f to obtain respectively yg and у/ at the time instant tn+1.

As far as the time integration of system (31) is concerned, the time step (see the last equation in Section 3) has to account for the maximum acceleration over both body nodes and fluid particles, namely |a| = max(||a/j|oo, ||аь||оо)-

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