Fluid-Structure Interaction Analysis of Hydrofoils in a Pulsating Flow

The reduction of noise and vibration are very important in the design of hydrofoils. The current study focuses on establishing a theoretical and numerical model to investigate fluid-structure interaction caused by elastic hydrofoils in a pulsating flow. A fully coupled three dimensional boun dary element method (BEM) and finite element method (FEM) code is applied to analyze the hydrodynamic performance. The numerical results show that the peak frequencies of the support reactions are related to the natural frequency of the hydrofoil. The natural frequencies an d support reaction amplitudes are reduced significantly by including the fluid-structure coupling.


Introduction 1.Previous work
Flu id-structure interaction has become an important research area in the field of underwater structures and vehicles.Although determining the hydrodynamic loads acting on an elastic hydrofoil is still quite challenging, a number of experimental and numerical studies have been devoted to this topic.Jiho You et al. [1] and Suzanne [2] investigated the performance of elastic hydrofoil by CFD.Young [3,4] and Lin et al. [5] used potential flow theory to predict the coupled flu id-structural response of elastic propellers under cavitating and non-cavitating conditions.Liu et al. [6] and Torre et al. [7] conducted an experimental study into the hydrodynamic characteristics of an oscillating hydrofoil.

Objective of the present work
The present work aims to investigate the fluid-structure interaction of a hydrofoil in a pulsating flow.The methodology is presented in Section 2, hydrodynamic performances are given in Section 3 and the main findings are summarised in Section 4.

Structural model
A hydrofoil with one end free and the other end fixed is considered in this model.By using classical finite element theory, the d iscrete equation of motion for the hydrofoil can be written as: where i ' i ' , i ' i ' and i ' are the nodal acceleration, velocity, and displacement vectors, respectively.> @ M , > @ C and > @ K are the global structure mass, damp ing, and stiffness matrices.w F is the hydrodynamic force vector generated by the pulsating flow.

Pulsating flow
The pulsating flo w is generated by assigning the following time-dependent velocity 0 , , , , , cos where, 0 V represents the uniform co mponent of the inflow.

, , p x y z A
and Z are respectively the amp litude and frequency of the pulsating flow., , x y z are spatial coordinates, and t denotes time.

Hydrodynamic model
In order to calculate the hydrodynamic force w F , a BEM method is used to solve the boundary-value problems given by: 2 0 in Here, I is the hydrofoil-induced perturbation potential, n is the outward unit norma l vector, δ is the displacement vector of nodes on the hydrofoil

Hydrodynamic force
The perturbation potential, I , obtained in the preceding section is applied to the linearized Bernoulli equation : to predict the pressure fluctuation on the hydrofoil surface.These hydrodynamic forces are then decomposed into two parts : Here, ' , and can be expanded as F has no relationship with the structure and provides excitation for the fluid-structure coupling system.

Governing equation
By moving F to the left side of equation ( 1), the governing equation becomes > @ > @ > @ > @ > @ > @ > @ > @ > @ > i @ (8) By Appling a FEM code co mbined with a BEM code, with the dynamic response predicted using Wilson-theta method in classical vibration theory.

Symmetric hydrofoil
The co mbined BEM [8, 9, 11-13] and FEM method has been used to calculate a NACA 0015 hydrofoil (Fig. 1) moving in a pulsating flow with various frequencies given by n Z S , 2,4,...,28,30 n .The hydrofoil has one end fixed (shown in b lue in Fig. 1) and the other end free.The chord length is 1 m and its span length is 4 m.The density, Poisson's ratio and elastic modulus of the hydrofoil are 7800 kg/ m 3 , 0.3 and 2.1x10 11 Pa, respectively.In this work, the damping matrix of the hydrofoil is neglected.The inco ming uniform velocity is 10m/s, angle of attack is 10 D , and amplitude of the pulsating flow is 0.1 m/s in the z direction.The first natural frequency of the hydrofoil is 7.71Hz without considering the flu id-structure coupling matrices and 5.59Hz when those matrices are considered.As the support reactions in the x and y directions are very small, only the support reactions in the z d irection are presented.Figure 2 shows the impact of the flu id-structure coupling matrices on the react ion force amp litudes as a function of pulsating flow frequency.

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Figure 2 shows that: 1.The maximu m reaction force occurs when the pulsating flow frequency matches the first natural frequency of the hydrofoil.This is true regardless of whether the flu id-structure coupling mat rices are included.There is no contradiction with classical vibration theory.
2. Including the fluid-structure coupling matrices reduces the first natural frequency by approximately 27.5%.As Figure 3 shows, the other natural frequencies are also considerably reduced, by including the fluid-structure coupling matrices.
3. The amplitude of the support reaction at the first natural frequency is 36% less when the fluidstructure coupling matrices are considered.

Asymmetric hydrofoil
For co mparison, the combined codes are applied to an asymmetric NACA 0015 hydrofoil (Fig. 4), with a mean camber line given by   1.No contradiction has been found with the classical vibration theory.
2. The natural frequencies decrease when the fluidstructure coupling matrices are considered.
3. The amplitude of the support reaction at the first natural frequency is less when the fluid-structure coupling matrices are considered.

Conclusions
This paper has presented a theoretical and numerical method to study fluid-structure interaction of hydrofoils in a pulsating flow.Both symmetric and asymmetric hydrofoils have been investigated by the combined codes.
Results reveal that: 1) natural frequencies of hydrofoils have a significant reduction when the effects of fluidstructure coupling are considered.2) the amplitude of the support reaction at the 1st natural frequency is approximately 33% less when the fluid-structure coupling matrices are considered.3) flu id-structure interaction plays an important role in the performance of elastic hydrofoils, even though the elastic deformat ions are small.
DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, 201 ju mp across the hydrofoil surface at trailing edge.tc denotes the time required for the flu id to travel along the wake surface fro m the hydrofoil trailing edge re R to the wake point wake R .The Laplace equation is comb ined with the Morina-Kutta condition to determine the hydrofoil-induced perturbation potential, and kinematic boundary conditions under the small deformation hypothesis ( see more details in[3,4, 10]  ).

K
are the added-mass, added- damping and added-stiffness matrices.Meanwhile, 2 w

Figure 2 .
Figure 2. Reaction amplitudes for varying pulsating flow frequency for the symmetric NACA0015 hydrofoil.

Figure 3 .
Figure 3. Natural frequency comparison for the symmetric NACA 0015 hydrofoil.
rat io of sagitta to chord length is 0.1.Figures5 and 6show the results of this case.The obtained results show that:

Figure 6 .
Figure 6.Natural frequency comparison for the asymmetric hydrofoil.