## 3 Materials and Methods
### Computational Fluid Dynamics  
The Jellyfish displacement is produced by its propulsion system generated by contractions. When these happen, their body generates a vortex, which is forcefully expelled, and produces a thrust that generates a reaction that drives it forward. To keep the movement constant, a series of pulsations must be generated. To validate the geometry model, all forces involved in the displacement of a body in the water, such as drag coefficient, added-mass coefficient and Reynolds number, are analyzed [50]. Together with the data obtained from the CFD, they allow evaluation of the instantaneous acceleration produced by the contraction and determine if jellyfish can advance [45]. During the development of the simulation, gravity has been used as an active parameter, because its use implies the action of buoyancy. Figure 8a,b shows the fluid displacement (water) during the vertical and lateral contraction respectively represented by spheres, where there is a variation in the sphere color according to the velocity generated during the process, the spheres being red for high-speed areas and blue for negative velocity areas. The accelerations are shown with lines in Figure 8c,d for vertical and lateral contraction, respectively. Figure 8: Representation of fluid displacement during jellyfish contraction using the Solidworks-Flow Simulation plug-in: (**a**) vertical displacement—speed (m/s); (**b**) lateral displacement—speed (m/s); (**c**) vertical Acceleration (cm/s\({}^{2}\)); (**d**) lateral acceleration. Source: Author. Figure 7: Diameter of the jellyfish bell variation comparison. The verification of simulation is performed based on Equation (2) [51], which evaluates the average acceleration in order to determine if a positive displacement is generated. This expression involves the water jet thrust under the hood (\(T_{i}\)) [g\(\cdot\)cm/s\({}^{2}\)], the drag force (\(D_{i}\)) [g\(\cdot\)cm/s\({}^{2}\)], the weight (\(W\)) [N] and the effective mass of the jellyfish (\(E_{i}\)) [g]. This equation does not consider the presence of tentacles, because of the complexity of modeling for them; for that reason, a known model was considered to represent the most relevant part of the body (around 92% of the body is modeled as a bell shape). The calculation without including the tentacles shows that there will be a positive displacement, while the simulation corroborates that tentacles generate an additional impulse because of the positive vortices generated at the ends after each contraction. In this study, the robot presents a slow motion, and the positive feedback on thrust associated with \"added-mass variation\" is very small and has not been considered [52]. \[A_{m}=\frac{\left[T_{i}-\left(D_{i}+W\right)\right]}{E_{i}} \tag{2}\]  
which are calculated, respectively, from the expressions: Equation (3) for (\(T_{i}\)), Equation (4) for (\(D_{i}\)), Equation (5) for (\(E_{i}\)) and Equation (6) for (\(\alpha_{i}\)) [53; 54; 55; 51]. \[T_{i}=\left(A_{v}\right)\alpha_{i}j_{i}^{2} \tag{3}\]  
\[D_{i}=\frac{1}{2}\rho.u_{i}^{2}.(\pi.h_{i}.\frac{d_{i}}{4}).C_{D,i} \tag{4}\]  
\[E_{i}=V_{b,i}.\rho.[1+\alpha] \tag{5}\]  
\[\alpha_{i}=0.5\frac{d_{i}}{\left(h_{i}\right)^{1.4}} \tag{6}\]  
The variables involved are bell hole area (\(A_{v}\)) [cm\({}^{2}\)], water density at 25 \({}^{\circ}\)C (\(\rho\)) [g/cm\({}^{3}\)], fluid velocity (\(j_{i}\)) [cm/s] instant jellyfish speed (\(u_{i}\)) [cm/s] jellyfish bell height (\(h_{i}\)) [cm], jellyfish diameter (\(d_{i}\)) [cm], drag coefficient (\(C_{D,i}\)), volume inside the jellyfish bell (\(V_{b,i}\)) [cm\({}^{3}\)] and the added mass (\(\alpha\)). Equations (7)-(9) were used to calculate the previous values:  
\[j_{i}=\left(\frac{1}{A_{v}}\right)\left(\frac{dV_{s,i}}{dt}\right) \tag{7}\]  
For Equation (3), where \(j_{i}\) [cm/s], is the jet speed at time i, \(A_{v}\) [cm\({}^{2}\)] is the umbrella's whole area and \(V_{s,i}\) [cm\({}^{3}\)] is the instantaneous volume under the umbrella.