Creación de una Unidad Ejecutora en la Dirección de

The location of the free surface relative to the centre of the SPH particles can be determined if the particles are assumed to be optimally spaced. This provides a further check on the sensibility of the ‘guidelines’ developed earlier.

4.9.1 Two dimensions

For closely packed SPH particles in two dimensions the exact location of the free surface can be found from a geometric study.

Figure 4-21 shows three SPH particles on the surface. The free surface will be at a distance of hc from the centre of the particle circle when the areas A1 and A2 are equal. At this condition there is an equal “area” of the fluid (in two dimensions) above the free surface but within the particle as there is a void beneath the free surface between adjacent particles.

Figure 4-21. Location of the free surface in two dimensions.

For this geometry, R is the radius, θ the angle of the segment, d the depth of the segment. The area of the arc segment A2 is given by:

2 ோ

మ

ଶ (29)

And the angle, θ_{, is :}

2 arccos௛೎

ோ (30)

The area A1 is the area of the rectangle (2Rhc) less the area of the semicircle without the

segment: 1 2௖ ଵ ଶ ଶ 2 (31)

Figure 4-22 shows the curve of areas A1 and A2 as a function of the distance h. The crossover point is at 0.785. This implies that the contact thickness should be 0.785R.

R

θ

Free Surface Area A2

Area A1 hc d

B K Cartwright, 2012. p.52 The packing density of circles in a plane is only 90.69%, i.e. the voids account for almost

10% of the area. The volume associated numerically to the SPH particles is independent of the spacing and so the volume of each SPH particle should be larger than if they just touch, to account for the volume of the voids.

Figure 4-22. Areas of 2D SPH particle above the free surface (A1) and area of void below the free surface (A2) for contact thickness h, assuming an SPH volume based on the centre-centre distances.

Following the above process to find the waterline, but now considering that the area of each SPH particle is 1/0.9069 larger, the cross over point changes to 0.825 of the radius, as shown in Figure 4-23.

Figure 4-23. Location of the free surface in 2D accounting for the voids between the particles. The surface is located distance h from the particle centres where the two curves cross over.

Expressed as a ratio of the SPH spacing, the theoretical value of hc/s for the location of the waterline is 0.825 / (2R) = 0.4125. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 A2 A1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 A2 A1 Contact Thickness, hc (m) Area, m2 Contact Thickness, hc (m) Area, m2

B K Cartwright, 2012. p.53 This theoretical value does not agree with the observed experimental value of approximately 0.6 presented in Figure 4-4. The reason for this is likely to be the assumption of perfect packing for the theoretical solution, whereas in practice, the packing of the SPH particles is not likely to be uniform and optimal, particularly adjacent to the submerged object, as explained earlier. The packing cannot be greater than the theoretical maximum density, and so any other distribution of SPH particles will result in a lower local density, implying a larger contact distance will be required.

Extending this to the free surface, a similar effect can be expected. Consequently it will be difficult to identify exactly where the free surface is, but it should be possible to define a range where it will lie, based on the spacing of the particles.

4.9.2 Three dimensions

A similar study can be made in three dimensions. The definition of the volumes is shown in Figure 4-24.

Figure 4-24. Dimensions and volumes for location of the free surface in three dimensions.

Following the concepts for 2D, Figure 4-25 shows a curve of the volume of the spherical cap and the volume of the void below the line that is hc away from the centre. The cross over point indicates where the free surface should lie, in this case at a distance of 0.605R from the centre of the SPH particles.

3D packed spherical particles have a volume ratio of 74.05%, so the actual volume of the particles must be greater than the spacing would allow for touching spheres with a diameter equal to the spacing. The actual volume of each particle should be increased by (1/0.7405) to ensure that the voids are filled. Using a particle radius based on this larger volume, the curve of Figure 4-26 indicates that the free surface will be located at 0.817R from the particle centres.

Expressed as a ratio of the particle spacing, the theoretical values of hc/s is 0.408 in 3D.

B K Cartwright, 2012. p.54

Figure 4-25. The free surface in 3D is located at a distance away from the cente of the SPH particle equal to the cross over point of the two curves, when considering a volume based on the particle spacing.

Figure 4-26. The free surface in 3D is located at a distance away from the centre of the SPH particle equal to the cross over point of the two curves, when considering particle volumes accounting for voids in the

packing.