KEY CONCEPTS
• A class of marine mammals, known as cetaceans, use their tails as airplane hydrofoils to swim through the water.
• A new model has now been developed to predict the swimming efficiency of cetaceans.
• The propulsion efficiency of a specific tail shape is tailored to its swimming kinematics. 

As better and smarter robotic technology becomes available through the implementation of artificial intelligence, the possibilities improve for developing ways to use them in fluid environments, whether they be as underwater vehicles or helping to real-time monitor lubricant systems. Finding techniques for developing these robots so they operate in an efficient manner is the challenge.

In a previous TLT article,¹ researchers powered a lionfish-inspired robot using a redox flow battery. The robot used the battery as an electrolyte hydraulic fluid system to power pumps that moved its pectoral fin and tail. Experiments were conducted in a fish tank on the robot, which consisted almost entirely of fluid (approximately 90% by volume).

More efficient swimming through water is achieved by a class of marine mammals known as cetaceans. Examples include dolphins, whales and porpoises.

Keith Moored, assistant professor of mechanical engineering and mechanics at Lehigh University in Bethlehem, Pa., says, “Cetaceans are able to use a combination of morphological and kinematic characteristics to efficiently move through water. Morphology involves examination of the shapes of the flukes used by the cetaceans as they swim (see Figure 1). Kinematics refer to the motion of the cetaceans, including how each species oscillates its fluke in a specific way to maximize efficiency.”


Figure 1. The shape of flukes such as the one shown is an important characteristic influencing the swimming efficiency of cetaceans. Figure courtesy of Lehigh University.

Cetaceans use a predominantly circulatory-based propulsion method with their flukes to traverse through the water. Moored says, “These animals swim in a manner similar to how an airplane flies. Their tails act as hydrofoils in oscillating up and down. A circulatory water flow is produced around their flukes, generating lift in an analogous manner to how air flows over an aircraft wing.”

The morpho-kinematic combinations developed by cetaceans are dependent on how these marine mammals live. Moored says, “Some maximize efficiency because they migrate from one region to another over long distances where speed is not the primary consideration. Other cetaceans live in local areas and rely more on speed where their fins need to generate force quickly. These two needs represent competing factors.”

There has not been a model available to predict the swimming efficiency of a cetacean based on an evaluation of its fluke. In an effort to better understand how cetaceans propel through water, Moored and his colleagues have now developed a model to predict this parameter and to help explain why cetaceans are efficient swimmers.

Aspect ratio
The researchers examined fluke shapes from the beluga whale, bottlenose dolphin, killer whale, spotted dolphin and the false killer whale. In assessing swimming efficiency, the key parameter evaluated was the aspect ratio. Moored says, “When viewed from above, the aspect ratio is a measurement of the ratio of the span length of the fluke to its chord length. Essentially, long thin hydrofoils have a high aspect ratio, while short, stubby ones have a low aspect ratio.”

The researchers inputted shape and kinematic data for the five species into their numerical solver and then ran simulations. Propulsion efficiencies for all five species were accurately predicted by the model. Moored says, “We found that the propulsive efficiency of the specific shape of a cetacean fluke is tailored to its swimming kinematics. At first our simulations did not show this, but after developing our scaling model, we were able to establish the relationship between shape and swimming kinematics.”

All five species achieved swimming efficiency values from the model that were above 80%. The researchers found that the most efficient swimmer is the false killer whale, and the swimmer with the fastest relative speed is the spotted dolphin. Moored says, “We were not very surprised by the results from the five cetacean species. Even the beluga whale, whose morphology does not suggest its efficiency will be similar to the other cetaceans, displayed a very efficient rating from our model.”

The researchers also evaluated the fluke shape of one cetacean with the kinematics of another one to look for trends. In running 25 of these simulations, the researchers found that the most efficient fluke was from the pseudo orca fin shape, and the most efficient kinematics was obtained from the beluga whale no matter the fluke used.

One other parameter that impacted the model was the added mass forces attributed to the water surrounding the cetacean. Moored says, “We found that the added mass forces acting on the fluke play an important role in predicting its swimming efficiency. As the fluke is oscillated, the added mass forces generated from the acceleration of the water surrounding the fluke.”

A common example of the generation of added mass forces is the process of an individual making a “belly flop” upon entering the water. Moored says, “A large added mass force acts on a person’s body when there is a sudden acceleration of fluid and a large contact area, which can be quite painful. In contrast, diving into the water produces less contact area, which leads to the acceleration of a smaller volume of water and a smaller added mass force.”

The model is able to predict swimming efficiency, whether a fluke is oscillating at a frequency of one hertz or five hertz. Moored says, “We are working to refine the model by incorporating other secondary effects that—while not as important as the aspect ratio—might still influence swimming efficiency.”

The researchers hope this model will help in the development of underwater robots. Moored says, “We are working in the field of biomimetics where we try to use bio-inspired engineering in developing new devices. Our intent is to use this model in that manner but not to necessarily replicate the shape and kinematics of any particular cetacean.”

Additional information can be found in a recent article² or by contacting Moored at kmoored@lehigh.edu.

REFERENCES
1. Canter, N. (2019), “Robot powered by hydraulic/energy storage system,” TLT, 75 (10), pp. 12-13.
2. Ayancik, F., Fish, F. and Moored, K. (2020), “Three-dimensional scaling laws of cetacean propulsion characterize the hydrodynamic interplay of flukes’ shape and kinematics,” Journal of the Royal Society Interface, 17 (163). Available here.