# Study on Oscillatory and Undulatory Motion of Robotic Fish
## 3 Results and Discussion
### Robotic Fish with Oscillatory and Undulatory Motion  
The first simulation involves oscillation in two primary directions: investigating variations in flapping frequency and its interaction with the flow. During each up or down cycle of the model, resistance is generated across the entire wetted surface, resulting in rearward thrust opposing the direction of motion. It is crucial to emphasize that the drag coefficient of a particular foil depends not only on its profile shape but also on its Reynolds number. Thus, maximizing thrust and efficiency at a specific Reynolds number presents an optimization challenge, requiring the identification of the model shape with the lowest drag coefficient for a given motion profile (as shown in Figure 13). Figure 12: The comparison of the drag coefficients between the results of theoretical and numerical methods. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **U** \\ **(m/s)** \\ \end{tabular} & \
\begin{tabular}{c} _Re\(\times\)10\({}^{6}\)_ \\ **(–)** \\ \end{tabular} & **Theoretical** & **Numerical** &
\begin{tabular}{c} **Difference between Numerical and** \\ \(\left|\text{{Conumerical}}-\text{{C}}_{\text{t}}\text{{theoretical}}\right|\times\)**100\%** \\ \end{tabular} \\ \hline
1.0 & 1.146 & 0.264 & 0.290 & 9.930 \\
1.5 & 1.720 & 0.260 & 0.282 & 8.297 \\
2.0 & 2.293 & 0.257 & 0.277 & 7.607 \\
2.5 & 2.866 & 0.255 & 0.274 & 7.097 \\
3.0 & 3.439 & 0.254 & 0.271 & 6.794 \\
3.6 & 4.127 & 0.251 & 0.268 & 6.544 \\
4.0 & 4.586 & 0.250 & 0.266 & 6.345 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between theoretical and numerical results of the drag coefficients for flow. The dorsal fin in ray-finned fish generates significant lateral momentum during locomotion, resulting in roll and yaw torques. To achieve stable swimming, other fins must counterbalance these torques. The tail fin may offset roll torques, and coordination between the tail and pectoral fins helps correct yaw torques. Theoretical research highlights the existence of a multifaceted force equilibrium among all fins, the body, and the tail, even during consistent swimming. This challenges the conventional classification of locomotion as either BCF (body-caudal fin) or MPF (median-paired fin), as median fins actively contribute to body stability (as shown in Figure 14). To comprehend this force equilibrium, data on buoyancy, body shape, and fin forces are crucial. Fish frequently maintain a positive angle of attack to their bodies, impacting lift forces due to body airflow. The three-dimensional flow dynamics around the body and tail introduce further complexity. Vortex wake patterns generated by different fins interact, creating a dynamic hydrodynamic environment. Comprehensive theoretical investigation is essential to fully grasp these interactions, underscoring the significance of three-dimensional effects in understanding fish undulatory hydrodynamics. At a flapping frequency of 2 Hz, it is observed that both types of motion experience a reduction in their respective drag coefficients, while the overall drag forces they encounter increase. Upon closer examination of each specific motion, it becomes apparent that the undulatory motion exhibits a higher drag coefficient when compared to the oscillatory motion. Conversely, the oscillatory motion demonstrates a lower drag force in comparison to the undulatory motion. These findings suggest that the dynamics of these two distinct types of motion play a crucial role in shaping their drag characteristics, with undulatory motion displaying a more pronounced resistance to flow and oscillatory motion exhibiting a relatively lower drag force (as shown in Figure 15 and Table 4). In contrast to the 2 Hz flapping frequency, there are distinct results observed at the 5 Hz flapping frequency. At 5 Hz, the drag coefficient is significantly higher compared to 2 Hz, although it still exhibits a decreasing trend. The drag force at 5 Hz is also greater than at 2 Hz, but there is a notable difference: it decreases as the flow velocity increases.