Finite element method has been used to analyze the propulsive efficiency of a swimming fin. Fluidstructure interaction model can be used to study the effects of added mass on the natural frequencies of a multilayer anisotropic fin oscillating in a compressible fluid. Water by neglecting viscidity effects has been considered as a surrounding fluid and the frequency response of the fin has been compared with that of vacuum conditions. It has been shown that because of the added mass effects in water environment, the natural frequencies of the fin decrease.
Multilayer anisotropic structure has wide applications in areas such as modern construction engineering, biomechanical engineering, aerospace industries, aircraft construction, and the components of nuclear power plants. It is therefore very important that the modal and dynamic analysis of multilayer anisotropic structure when subjected to different loading conditions be clearly understood so that they may be safely used in these industrial applications.
It is well known that the natural frequencies of structures in contact with fluid are different from those in vacuum. Therefore, the prediction of natural frequency changes due to the presence of the fluid is important for designing structures which are in contact with or immersed in fluid. In general, the effect of the fluid force on the structure is represented as added mass, which lowers the natural frequency of the structure from that which would be measured in a vacuum. This decrease in the natural frequency of the fluidstructure system is caused by increasing the kinetic energy of the coupled system without a corresponding increase in strain energy.
In this paper the propulsive efficiency of a swimming fin has been studied. Dynamic analysis of aquatic locomotion is a fundamental parameter in the performance search. In the case of swimming with fins, the propulsive efficiency depends on several factors. Most models suggested aim to evaluate the dynamic performances, including drag and lift which are the two relevant parameters relevant to quantifing the propulsive efficiency of a fin. Some proposed models are essentially of discrete type [
The numerical formulations used include the displacement formulation [
In this work, we assume an amateur swimmer, where the scale of velocity
Dimensional analysis of coupled equations (NavierStokes equations and governing equations of nonlinear elasticity) of a fluidstructure interaction model [
Geometry of 3D fin.
2D computational domain.
The use of the ALE method is not essential in this study because the material is assumed linear. In the frame attached to the fin, the problem is to find
solid domain
fluid domain
fluidsolid interaction
other boundary conditions:
Modal analysis of elastic submerged structures is needed in every modern construction and has wide engineering application especially in ocean engineering. In this study, modal analysis is important to predict the dynamic behavior of the submerged fin. It is well known that the natural frequencies of the submerged elastic structures are different from those in vacuum. The effect of fluid forces on the submerged fin is represented as added mass, which decreases the natural frequencies of the submerged fin from those which would be measured in the vacuum. This decrease in the natural frequencies of the submerged structures is caused by the increase of the kinetic energy of the fluidfin system without a corresponding increase in strain energy. This step seems important to calculate the variations of natural frequencies of the fin for different situations. For this, we looked at the modes of the fin in the vacuum and water. Indeed, to test the quality of a fin, it is usual to search its quasistatic deformed shape and dynamic response in air. The aim is to check if results of tests carried out of the water are strongly influenced by the presence of the surrounding fluid. In addition, frequencies can have accurate information in the dynamic behavior of the system. By introducing the spaces of test functions
Two types of calculations were carried out. The first is when the palm is plunged into the vacuum and the second into water. We give below the results for a model of up to five layers (
Natural frequencies
Oriented fibers  In vacuum  



 
0°/90°  12.6  76.0  205.3 
0°/90°/0°  29.3  122.9  265.0 
0°/90°/0°/90°  22.6  90.1  192.0 
0°/90°/0°/90°/0°  28.8  79.7  185.1 
Natural frequencies
Oriented fibers  In water  



 
0°/90°  1.7  46.0  103.1 
0°/90°/0°  3.5  21.7  55.1 
0°/90°/0°/90°  2.4  14.5  37.3 
0°/90°/0°/90°/0°  2.7  11.8  33.4 
Natural frequencies
Oriented fibers  In vacuum  



 
90°/0°  12.6  73.5  158.4 
90°/0°/90°  9.5  51.3  105.5 
90°/0°/90°/0°  22.1  56.2  115.5 
90°/0°/90°/0°/90°  17.4  44.8  91.6 
Natural frequencies
Oriented fibers  In a water  



 
90°/0°  1.7  13.3  39.4 
90°/0°/90°  1.1  8.7  23.8 
90°/0°/90°/0°  2.2  9.5  23.8 
90°/0°/90°/0°/90°  1.6  6.8  17.2 
The colors pertain to the pressure field and the arrows to the velocity field in the case of
Coupled mode 1 (
Coupled mode 2 (
The colors pertain to the pressure field and the arrows to the velocity field in the case of
Coupled mode 1 (
Coupled mode 2 (
The dynamic problem was conducted using the data proposed in [
To avoid a resonant frequency, the excitation frequency is taken far enough from the first natural frequency of the coupled system. The hydrodynamic parameters that seem most relevant are the total force
The rotation
Thrust of fin in the case of translation motion.
The function
In order to have a reasonable performance of the system, we must combine both translational and rotation motion and take the full expression of the excitation force
Finally, the above results allow us to draw some conclusions.
The presence of layers provides some flexibility as indicated by the results of modal analysis. The first mode is flexural type, which justifies the use of models proposed in [
Fins with anisotropic material structures allow implementing a technique of layers parameterization to improve performance. It is quite possible now to bring special attention to the structure of the layers and types of constituent materials thereof.
The sensitivity of the dynamic behavior of the model with respect to the materials used and the boundary conditions for the fluid domain should be noted. Indeed, the presence or absence of rigid walls alters significantly the natural modes of the coupled system. Thus, the dynamic behavior of a swimmer depends on the localization in the pool where it is at the given moment. To obtain a better thrust, the fin has to be elastic and has to be sought at least in rotation. The amplitude of the vertical translation must be controlled to avoid a too high lift, in order to remain at a constant depth. The use of multilayer fins allows controlling an excessive variation of lift (Figures
Most experimental results we know [
Lift of fin in the case of translation motion.
Thrust of fin in the case of rotation motion.
Lift of fin in the case of rotation motion.
Thrust of fin in the case of combined rotational and translational motion.
Lift of fin in the case of combined rotational and translational motion.
In this paper the modal and dynamic analysis is proposed to understand the behavior of a flexible composite fin with a good accuracy. The publications in the literatures deal with the behavior of fins; few authors have not studied the case of coupled boundary conditions [
The authors declare that there is no conflict of interests regarding the publication of this paper.