A wavelet immersed boundary method for two-variable coupled fluid-structure interactions

Highlights

• A wavelet IB method coupled a traditional IB method and a wavelet FEM.

• A boundary influence matrix is constructed to restrain the non-physical oscillations.

• A series of B-spline wavelet delta functions improve the interpolation accuracy.

• The numerical examples show that the new method can solve the two-variable coupled FSI problems and alleviate the non-physical oscillations efficiently.

Abstract

In this paper, a wavelet immersed boundary (IB) method is proposed to solve fluid-structure interaction (FSI) problems with two-variable coupling, in which it is an interaction between fluid force and boundary deformation. This wavelet IB method is developed by introducing a wavelet finite element method to calculate the FSI force affected by the two-variable coupling. Furthermore, a boundary influence matrix and a series of B-spline wavelet delta functions are constructed to restrain the non-physical force oscillations. Finally, several FSI problems are simulated, which include flows past a fixed circular cylinder and a crosswise oscillating circular cylinder, as well as an in-line oscillating circular cylinder in a rest fluid. The numerical examples show that the new method is a simple and efficient method for two-variable coupled FSI problems.

Introduction

FSI problems are common problems in the fields of mechanical engineering, aerospace and biodynamics, so many researchers have investigated many numerical methods to work out these problems. There are many computational fluid dynamics methods to compute fluid field variables, such as vortex-lattice methods, boundary element methods, finite volume methods, response surface methods, and so on. However, there is only one kind of method for calculating structure field variables and widely applying in engineering structure analysis, i.e. finite element methods (FEMs). Therefore, many researchers solve FSI problems by coupling one of the computational fluid dynamics methods and a FEM. Lin et al. [1,2] adopted the vortex-lattice method and FEM to analyse the FSI problem of propeller, and Kim et al. [3,4] made use of the boundary element method and FEM to figure out the springing phenomenon of a large ship. Moreover, based on the finite volume method and FEM, Maki et al. [5,6] investigated the hydro-elastic analysis of bodies that enter and exit water. When it comes to response surface method, Hosseinzadeh et al. [7,8] studied optimization of hybrid nanoparticles with mixture fluid flow in an octagonal porous medium. Whereas the above methods are based on an arbitrary Lagrangian-Eulerian idea, the grid processing mode is a fitted or conforming mesh method. These methods are troublesome when the structure is moved or deformed, since the new fluid and structure grids need to be reproduced for adapting the new fluid-structure boundary.

For overcoming this issue, some unfitted or non-conforming mesh methods are proposed, such as smoothed particle hydrodynamics methods, moving particle semi-implicit methods, IB methods, and so on. According to the idea of coupling one of the computational fluid dynamics methods and a FEM, Fourey et al. [9] employed the smoothed particle hydrodynamics method and FEM to simulate the violent FSI problem, and Yang et al. [10] applied the two method to study free-surface flow interactions with deformable structures. For the moving particle semi-implicit method and FEM, Mitsume et al. [11] developed a non-overlapping approach, and Hwang et al. [12,13] utilized a modified particle method to analyse sloshing flows.

As for IB method, it is an excellent and simple method to solve FSI problems, because the fluid-structure interaction is transformed into boundary force worked on fluid domain. However, there is a little literature on investigation of the coupling method between an IB method and a FEM. Generally, the IB methods can be classified two kinds of methods. On the one hand, the traditional IB method was firstly proposed by Peskin [14] to model the blood flow of heart valve, and Ghosh et al. [15] studied the drafting, kissing and tumbling process of two particles by using a traditional IB method. The advantages of these traditional IB methods are that the boundary deformation is calculated, and the algorithms of these methods are simple. However, for a large stiffness boundary, the large boundary deformation is acquired by interpolating the fluid velocity nearby IB directly, and it leads to an unreal boundary force. This is a reason that the boundary stiffness is large and the real fluid force acting on the immersed boundary is ignored. So, traditional IB methods are suitable for a soft immersed boundary FSI problems, but they are restricted by the time step and boundary stiffness for FSI problems with a large elasticity boundary.

On the other hand, Uhlmann et al. [16] developed a direct forcing IB method to satisfy the no-slip condition at a rigid boundary precisely, and Lo et al. [17] presented an efficient direct forcing IB method for fluid flow simulations with moving boundaries. The advantage of these direct forcing IB methods is that the fluid volume force is regarded as the FSI force acting on the fluid field, so the FSI force of the direct forcing methods can satisfy the no-slip condition more precisely than that of the traditional IB methods. Whereas, the deformation of the immersed boundary is neglected, and the algorithms of these methods are more complicated than those of the traditional IB methods, because of solving the external Poisson equation.

In order to consider the fluid force and the deformation of a large stiffness boundary, this paper introduced a wavelet finite element method to couple a traditional IB method. Compared with general FEM, wavelet finite element methods regard wavelet functions as interpolation functions to improve the computational precision of structural analysis. The wavelet functions include Daubechies wavelet [18], B-spline wavelet on the interval (BSWI) [19], Harr wavelet [20], [21], [22], [23], Second generate wavelet [24], Legendre wavelets [25], and so on. In all the above wavelet functions, Cohen [26] finds a conclusion that BSWI has the best approximation performance, and Goswami et al. [27] given a report that it can overcome the numerical oscillations on the boundary. Compared with the current wavelet interpolation functions, Xiang et al. [28,29] discovers the BSWI has the desired features of compact support, smoothness, symmetry and multiresolution analysis. Liu et al. [30] testified BSWI have the better performances of convergence and self-adaption, Vampa et al. [31] employed BSWI to analyse the boundary value problems, and Lakestani et al. [32,33] solved the nonlinear Fredholm-Hammerstein integral equations by using BSWI. Based on these research, Yang et al. [34,35] constructed a BSWI finite element of curved beam, which has excellent properties of analysing curved structure. In this paper, the BSWI finite element of curved beam is employed to depict the mechanical characteristics of an immersed cylinder boundary. Due to the superior properties of the BSWI, it is also applied to construct a series of B-spline delta functions, which can connect the fluid information on Eulerian grids and the boundary information on Lagrangian grids. Moreover, this paper made use of the different order B-spline delta functions and the boundary influence matrix, so as to restrain the non-physical force oscillations effectively.

Therefore, this paper proposes a wavelet IB method, by considering the boundary influence matrix, the two-variable coupling between the fluid force and the boundary deformation, and the B-spline delta function. First, the fluid velocity of a thin shell fluid around an immersed boundary can be called as fluid concentration velocity, and it is calculated by the fluid velocity nearby the immersed boundary, the B-spline delta function and the boundary influence matrix. Second, the fluid force is obtained by the fluid concentration velocity correction, and the boundary deformation is computed by the fluid force and the stiffness matrix of BSWI finite element. Finally, the two-variable coupled FSI force is computed by applying the two variables, and it is introduced into the governing equations of fluid field for accomplishing the two-variable coupled fluid-structure interactions. The aim of this paper is to accomplish the coupling between the BSWI finite element method and the traditional IB method, so as to analyse the FSI problems with a large stiffness boundary more precisely.

This paper is organized as follows. In Section 2, a wavelet IB method is proposed, including the incompressible fluid governing equations, fluid concentration velocity correction, fluid-structure interaction force density, B-spline wavelet delta functions, and a numerical scheme with formally second-order accuracy. In Section 3, several numerical examples are presented to validate the precision and efficiency of the wavelet IB method. Finally, some conclusions are given in Section 4.