Biomechanics and Medicine in Swimming XI

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¾ ¾ ¾ Methods A three-dimensional model of the fin-water system is proposed which is built on the following assumptions. Based on continuum theory, a fluid-structure interaction is developed with inertial coupling. The system is depicted in Figure 1 and is composed of a solid domain (ΩS ) coupled with a surrounding fluid domain (ΩF ). The boundary (Γ) ¾ corresponds to the deformable fluid-structure interface. The external excitation for the stroke motion (i.e. the rate of undulation) ¾ is assumed to be lower than the wave speed ¾ in the fin, so that the fluid can be ¾ de- ¾ scribed with the acoustic pressure (Morand and Ohayon, 1995). It corresponds to an undulatory type of low frequency swimming. The fin is assumed to be a deformable elastic structure. Namely, the monofin is modelled using a multilayer structure, whose constitutive law is written by means of the Cauchy stress tensorσ . The imposed kinematics at the leading edge of the monofin (that corresponds to the swimmer’s ankle) results in a volume force γ ie. If we denote by u the displacement field of the monofin and p pressure field within the fluid domain, the problem consists in finding (u,p) solutions of the following three-dimensional ¾ boundary value problem: ¾ r ∂ 2 u ∂t 2 u = = divσ + γ ie 0 (ΩS ) (Γ0 ) (1) (2) σ.n = −pn (Γ) (3) ∆p − 1 2 c0 ∂ 2 p ∂t 2 = div(−r 0γ ie) (Ω F ) (4) ∂p ∂ = −r0 ∂n 2 u ∂t 2 .n − r0γ ie.n (Γ1) (5) ∂p = −r0γ ie.n (Γ) (6) ∂n ¾ p = 0 (Γg ∪ Γs ∪ Γe) (7) xω(t) + zω γ ie = 2 (t) 0 zω(t) − xω 2 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎢ (t) ⎦ ⎥ Where c is the wave velocity in the fluid, r is the density of the mono- 0 ¾ fin, r is the fluid density and n is the unit outward normal to the fin 0 domain. The coupling results in a new term in the inertial contribution of the system. Indeed, this added mass characterizes the energy transmission ¾ between the fin and the water. The specificity of the present method is that ¾ the temporal evolution ¾ of dynamical features is available. Namely, transient added mass, three-dimensional forces and torques at the point corresponding to the swimmer’s ankle as well as energy balance are obtained. Since this resulting fluid-structure interaction problem is too complex to be solved analytically, we analyzed using the means of numerical approximation. We will briefly outline the solution methodology used to solve the governing equations. The latter are solved with the Finite Element Method using the commercial software COMSOL Multiphysics (Comsol Inc., Burlington, USA) . Numerical simulations are carried out for the fluid flow in a pool with the following dimensions: In the pool, a fin with realistic dimensions (0.80 m chord and 0.50 wide) is located at the distance of 5 m from left wall. In this study a monolithic approach is used. Indeed, the equations governing the fluid flow and the displacement of the structure are solved simultaneously with single solver. The main steps of the methodology are the following: first, based on realistic pictures, the geometry of the monofin is created. The resulting volume, which is the form of an Initial Graphics Exchange Specification (IGES) file format, is then input into the commercial software COMSOL Multiphysics and transformed to an unstructured volume mesh with triangular elements. The second step consists in the meshing. The final volume mesh (Figure 1) consists of 6256 triangular elements for the fin and 128676 triangular elements for the water. The third step consists in a modal analysis of the three-dimensional coupled problem. γ ie is set to zero and a time harmonic solution of the problem is computed. This allows obtaining the fundamental vibration characteristics of the immersed monofin and therefore the quantitative and qualitative effect of the added mass can be pointed out. The final step of the ¾ methodology consists in a transient analysis of propulsive forces. Thus, in the numerical experiments that are presented, the monofin is undergoing time-dependent motion that combines translating and rotating motions as follows chaPter2.Biomechanics ω(t) = θ0 sin(ω 0t +ψ) h(t) = h0 sin(ω 0t) with ω0 =1.7rad.s −1 θ0 = π rad 4 π ψ = 2 rad h0 = 3 4 c in which c is the chord of the monofin. In the numerical experiments, c is set to 0.7m. The material properties used in the FEM simulation are given in Table 1. Table 1. Material characteristics used in the FEM simulation. E is the Young’s modulus, ν is the Poisson’s coefficient and r is the density of the monofin. (Data provided by Breier SAS company) ¾ E [Pa] ν r [kg.m ¾ ¾ ¾ ¾ ¾ ¾ ¾ −3 ] Carbon 12K 146.10 ¾ 9 0.32 1565 Low density Polyethylene 146.10 ¾ ¾ ¾ 9 0.38 920 Rubber 0,05.10 ¾ ¾ ¾ 9 0.49 1250 Some of the key parameters associated with the performance of the monofin are the lift force L, the thrust ¾ force ¾ T (equal to negative drag force D) and the resulting torque at the swimmer’s. Thanks to the present FEM method, the accurate efforts (resulting force and resulting torque) at the point O, which corresponds to the swimmer ankle, are computed. The instantaneous resultant force components are calculated by directly integrating the traction vector on the fin surface. In particular, if σ is the stress tensor and n is the outward normal unit to the fin surface A, the resultant force is given by R(t) = ∫ σ.ndΓ = L(t).x + L(t).z A where T(t) and L(t) are the instantaneous thrust force and instantaneous lift force respectively. It known that the performance of flexible fins depends largely on the effective dynamic properties of the structure. Therefore, ¾ modal analysis was performed in this work. This allowed for ¾ the quantification of the added mass effect on the fin by comparing the natural frequencies and modal shapes in the case where the fin is submerged or not. The first natural frequencies fi = λi /2π are reported in Table 2. In this study transient force and torque at the swimmer’s ankle is computed. Figure 3 shows the variation of the thrust. Table 2. Comparison of first natural frequencies fi = λi /2π [Hz] obtained with the Coupled FEM. ¾ f1 f2 f3 2D in water 6.08 41.12 70.03 3D in vacuum 13.01 ¾ 113.51 256.34 3D in water 4.395 23.87 41.87 53
BiomechanicsandmedicineinswimmingXi Figure 2. Plot of the third modal shape for the monofin blade with the von Mises stress (top) and plot of the fluid pressure at time 1.5s of the transient analysis (bottom). There is a significant difference between the modal shapes of rigid fin model and that of flexible fin model. This highlights the necessity of accounting as accurately as possible not only the fluid-structure interaction but also the deformability of the elements during a propulsion analysis of a fin. Modal shapes of the immersed fin are not the same as those of the dry fin case. This is due to the non-symmetry of the added mass matrix. Therefore, including the effect of coupling between the fin and the surrounding water is essential for an accurate description of the dynamic behaviour of flexible fins. Moreover, this feature is an important parameter in the design of optimal swim fin especially for competitive swimmers. If we compare the eigenfrequencies obtained with the present model with those obtained with the corresponding two-dimensional model (Bideau et al., 2009), it can be observed that taking into account for three-dimensional effects leads to a decrease of the deformation of the fin. This can be explained by the fact that the three-dimensional model is less constrained that the two-dimensional one, which allows more flexibility in the third direction. Indeed, two-dimensional shapes can only take into account for the in-plane flexion. As an example, the third modal shape of the monofin blade is depicted on figure 2. On the same figure, the pressure field of the fluid is plotted at time t=1.5s. In fact, the pressure is being examined to better understand the generation of thrust. The temporal evolution of those variables tends to an oscillating behaviour when increasing the rigidity of the fin plate. It can be noticed that there is a significant variation of the frequency of the thrust evolution in regards to the applied stroke kinematics that are harmonically varying. This can be explained by the elasticity of the flexible fin. The elastic energy is stored in a first time and distributed in a second part. Therefore, the strain energy of the structure is seen to greatly affect the propulsive characteristics (propulsive force and resulting moment) of the monofin (Figure 3). Figure 3. Temporal evolution of thrust and lift for different material characteristics. 54 conclusIon A 3D fluid-interaction model has been presented. It is based monolithic resolution of the whole system fluid and solid. The present work investigated the properties of propulsive force of a three-dimensional deformable monofin. In particular, the added mass effect of the coupled dynamics has been pointed out. The transient propulsive force end torque computation has been investigated by means of FEM method. The influence of the elasticity (i.e. strain energy) has been investigated and shows that he flexibility of the fin blade increase the thrust. To our knowledge it is the first fluid-structure interaction model for the whole monofin-water system solved by 3D FEM approach. reFerences Bideau B. , Colobert B, Nicolas G, Fusco N, Cretual A, Multon F, Delamarche P (2003). How to compute the mechanical properties of monofins? In Chatard J.C. (Ed), Biomechanics and Medicine in Swimming IX, p.505-511, St Etienne. Bideau N, Mahiou B, Monier L, Razafimahery F, Bideau B, Nicolas G, Razafimahery F, Rakotomanana L (2009). 2D dynamical efficiency of a swimfin : a fluid-structure approach. Computer Methods in Biomechanics and Biomedical Engineering, 11, 53-54. Morand H and Ohayon R (1995). Fluid-Structure Interaction. John Wiley & Sons. Rejman M (1999). Dynamic criteria for description of single fin technique. In K.L. Keskinen, P.V. Komi and P. Hollander (Eds) Biomechanics and Medicine in Swimming VIII, p.171-176, Jyväskylä. Von Loebbecke A, Mittal R, Mark R, Hahn J (2009). A computational method for analysis of underwater dolphin kick hydrodymamics in human swimming. Sports Biomechanics, 8, 60-77. Zamparo P, Pendergast DR, Termin B, Minetti AE (2005) Economy and efficiency of swimming at the surface with fins of different size and stiffness. European Journal of Applied Physiology, 96, 459-470.
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