Journal of Engineering

Volume 2014, Article ID 541953, 8 pages

http://dx.doi.org/10.1155/2014/541953

## Fluid-Structure Interaction Effects on the Propulsion of an Flexible Composite Monofin

^{1}Arts et Métiers ParisTech, ENSAM Angers, 2 boulevard du Ronceray, 49035 Angers, France^{2}IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Received 17 May 2014; Revised 17 November 2014; Accepted 26 November 2014; Published 14 December 2014

Academic Editor: M’hamed Souli

Copyright © 2014 Adil El Baroudi and Fulgence Razafimahery. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Finite element method has been used to analyze the propulsive efficiency of a swimming fin. Fluid-structure 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.

#### 1. Introduction

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 fluid-structure 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 [1, 2], while others, by being inspired by organs of propulsion of marine cetaceans, use continuous models [3, 4]. Most of these authors do not account for the highly coupled nature of the problem. In fact, for rate of stresses observed in actual swimming, the coupling between the fluid and the fin becomes stronger.

#### 2. Governing Equations

The numerical formulations used include the displacement formulation [5], the potential formulation [6], the pressure formulation [7], and the combination of some of them [8]. Finite element method is used to extract the natural frequencies and modal shapes. To compute the natural vibration modes of a fluid alone, the fluid is typically described either by pressure or by displacement potential variables. When the fluid is coupled with a solid, standard methods to solve (1) and (2) consist in eliminating either the pressure or the displacement potential [9]. However, in both cases nonsymmetric eigenvalue problems are obtained (see, e.g., [10]). To avoid this drawback, Morand and Ohayon introduce in [6] an alternative procedure which consists in using pressure and displacement potential simultaneously. In this section we summarize their approach; further details and discussions can be found in their book [11].

In this work, we assume an amateur swimmer, where the scale of velocity is supposed to be very small compared to the compression wave velocities in the fin. Indeed, some amateur swimmers note that, when making foot movements at low frequency, the resonance phenomenon and buckling phenomena appear. And we cannot explain why these phenomena tend to occur, because the natural frequencies of the fin which would be measured in the vacuum are higher than of the beat frequency of an ankle, for example. In this study, we assume that the swimmer does not disturb the free surface of the fluid domain. This leads to neglect the gravity effects.

Dimensional analysis of coupled equations (Navier-Stokes equations and governing equations of nonlinear elasticity) of a fluid-structure interaction model [12] reveals several dimensionless parameters. One of its dimensionless parameters, , is called displacement parameter. The displacement parameter allows characterizing the nature of the coupling problem considered in this work. In the amateur swimmer hypothesis where , we can set the parameter to a very low value and we can show that the convective terms and viscosity terms can be neglected for the fluid model [12]. We can also assume the assumption of small deformations for the fin. The resulting model is called inertial coupling [13]. The real shape of the fin is given in Figure 1, but for the sake of the simplicity the problem is bidimensional (Figure 2) and the fin is immersed in a large pool. The fin is modeled by a multilayer linear elastic transverse anisotropic material. The different layers constituting the fin are denoted by and have the density . We denote by the displacement field in the fin and the pressure field in the fluid. and denote the sound celerity and density of the fluid, respectively. The longitudinal axis of the fin is denoted by . The force given in (1) is used to describe the motion of the fin. denotes the orientation of fibers relative to the longitudinal axis on the fin and takes or . Here, each layer is made of either fiberglass or carbon fiber.