## 4 Results and Discussion  
First of all, a comparison of the outflow trajectory is presented between the analytical solution of Equations (6) and (7) (in red in Figure 10) and the result of the SPH simulation. As seen in Figure 10, analytical curves are fairly well-followed by the outflow particles at certain \(v_{q}\) values. The fall of particles at \(x\)-axis position along all the \(y\)-axis was validated. This allowed us not only to confirm a quadratic outflow curvature of the SPH method on free surface using our simulation context, but also to confirm how an outflow target during CFS can possibly be determined and tracked. For all the following test and results, the same vessel's dimensions were used: \(B=L=0.191\) m. Unless indicated to be different, please also assume a liquid level height \(A=0.1591\) m.  
Because forces, velocities and positions of particles based in a discrete time are computed, speed calibration of the system for further experimentations is another element to consider for testing. It is crucial to synchronize the main simulation loop time with the real fluid speed. This adjustment is conducted by setting the number of system loop calls so that the simulated frequency of the free surface slosh approximates to theoretical slosh frequency in horizontal motion. For a squared vessel, a robust analytical solution to the free surface slosh frequency of the first oscillation component can be found in (13) (p. 59), and is shown in Equation (17), where \(Freq\) is the oscillation frequency in hertz of a free surface elevation cycle in horizontal motion, \(B\) is the vessel width and \(A\) is the fluid height at rest. It was mentioned earlier that the slosh frequency was \(\omega^{2}=\frac{g}{l_{1}}\) for the linear, pendulum-based model used; however, Equation (17) is more appropriate to use, since this computes the liquid height, that, at low heights, approximates better to the real oscillation values. Additionally, notice that in Equation (17) \(Freq\) is not a function of the slosh height, so the slosh induction method can compute its fundamental slosh motion regardless of the angle \(\theta\)--an ideal condition to perform real time synchronization in the simulator. \[Freq=\frac{1}{2\pi}\sqrt{\frac{g\pi}{B}\tan\hbar\left(\frac{A\pi}{B}\right)}. \tag{17}\]  
Figure 11 shows a complete free surface cycle for three different values of \(A\) in our simulation results. Equation (17) is used as reference to set the system time and find similar values for the experimental surface elevation cycle adjusting the SPH fluid forces. Figure 10: SPH outflows (grey) versus analytic curvatures (red). A narrow outflow to the (**left**). A broad outflow to the (**right**). Note that, as \(A\) increases, the difference between analytical and simulated values of _Freq_ barely diverges. These results confirm the maximum \(\Delta t\) choice to run real time simulations, although a smaller \(\Delta t\) would give more accurate results. As mentioned earlier, AFS without SIM is not processed. However, for that case, a center of rotation of \(\theta\) would have to equal the center of mass of the liquid area. This setup cancels the moment of inertia by having equal parts of liquid mass around the center of rotation. However, there should be a maximum value of the tilting angle velocity \(\dot{\theta}_{max}\) of vessel 1 at which this angle is limited to avoid spilling. Figure 12 shows the simulated values of the relation \((\theta/(27-\theta_{1}))\) between the angles of rotation (\(\theta\) of the vessel, and \(\theta_{1}\) of the slosh) versus the vessel frequency \(\dot{\theta}\). From a physics perspective, it is obvious that the free surface angle will remain horizontal, with its center of rotation equal to the center of mass, and with the vertical component of \(\ddot{\theta}\) below the gravity force. In Figure 12, a maximum finding of \(\dot{\theta}_{max}\) is given by our running simulations.