MING Ping-jian, ZHANG Wen-ping

School of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China, E-mail: pingjianming@hrbeu.edu.cn

NUMERICAL METHOD FOR MULTI-BODY FLUID INTERACTION BASED ON IMMERSED BOUNDARY METHOD*

MING Ping-jian, ZHANG Wen-ping

School of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China, E-mail: pingjianming@hrbeu.edu.cn

A Cartesian grid based on Immersed Boundary Method (IBM), proposed by the present authors, is extended to unstructured grids. The advantages of IBM and Body Fitted Grid (BFG) are taken to enhance the computation efficiency of the fluid structure interaction in a complex domain. There are many methods to generate the BFG, among which the unstructured grid method is the most popular. The concept of Volume Of Solid (VOS) is used to deal with the multi rigid body and fluid interaction. Each body surface is represented by a set of points which can be traced in an anti-clockwise order with the solid area on the left side of surface. An efficient Lagrange point tracking algorithm on the fixed grid is applied to search the moving boundary grid points. This method is verified by low Reynolds number flows in the range from =100 Re to 1 000 in the cavity with a moving lid. The results are in a good agreement with experimental data in literature. Finally, the flow past two moving cylinders is simulated to test the capability of the method.

fluid-structure interaction, Immersed Boundary Method (IBM), Volume Of Solid (VOS), unstructured grids

Introduction

The multi-body and fluid interaction can be widely found in marine and ocean engineering, and is a very important issue in Computation Fluid Dynamics (CFD). Numerical methods were developed as a complementary to experimental and analytical studies, including the Lagrangian moving mesh method[1]and the Chimera grid method[2,3]. The Lagrangian moving mesh is very inefficient or even impossible to simulate large amplitude motions, for which a remeshing procedure[4]is needed to deal with problems related with the low mesh quality. The Chimera grid method can deal with large amplitude moving body problems, but it is very expensive for the interpolation of variable values between moving grids and background grids. Recently, an immersed boundary method was proposed to simulate the fluid flow and rigid body interaction on fixed meshes. The immersed boundary method was originally proposed by Peskin in 1972 to analyze the blood flow in a human heart model. In his simulation, only a simple Cartesian mesh system wasused, with grids not necessarily conforming to the geometry of the heart[5]. The immersed boundary method is put into a mathematical formulation and the effects of solid boundaries are reflected in forces in the Navier-Stokes equations. There comes the socalled “Immersed Boundary (IB)” method, which gradually becomes a basic concept that the body is simply ‘immersed’ in the background mesh without necessarily fitting the underlying mesh points. Many modified IBMs were proposed, which are quite different from the most conventional CFD approaches[6]. More details of IBM history can be found in literature[7].

The Immersed Boundary Method (IBM) is originally implemented in the Cartesian grid, with an obvious drawback that the grid number is increased drastically with the increase of the Reynolds number[8]. In some situations, a great number of extra outside grids are required in the Cartesian grid, for example, in the lobe pump as shown in Fig.1(a). In order to reduce the extra grid number, IBM is extended to curvilinear grids to generate a body fitted grid through a coordinate transform[9]. In this article, the immersed boundary method is further improved, and then is used for unstructured grids as shown in Fig.1(b). The scopeof the immersed boundary method is thus extended. Combined with the concept of Volume Of Solid (VOS), this article focuses on the multi-body and fluid interaction problem.

Fig.1 Examples of computational grids

The immersed boundary methods can be roughly divided into two classes[10]. In the first class, the immersed solid surface is regarded as a “diffuse”interface of finite thickness to represent its effects on the surrounding fluid. The thickness is usually taken as the local mesh spacing. In this class includes most methods that employ forces and/or mass sources as well as those that use the volume-of-fluid and phasefield approaches. In the second class, the boundary is tracked as a sharp interface and the communication between the moving boundary and the flow solver is usually achieved directly through modifying the computational stencil near the immersed boundary.

In this article, the IBM method of the first class[11]is extended to deal with the multi-body and fluid interaction with unstructured grids, because it is easy to be used with only a few modifications of the exiting CFD codes. This method is implemented in an inhouse code to solve multi-body fluid interaction problems. The VOS can be further simplified by assuming that the body interior is occupied by the same incompressible fluid as that outside with a prescribed divergence-free velocity field. In this view, a fluidbody interface is similar to a fluid-fluid interface in the Volume Of Fluid (VOF) method for the two-fluid flow problems. The body can thus be identified by the VOS function as the VOF function[12,13]. Since the body is assumed to be incompressible, the pressure field inside the body obeys the same governing equation as that outside. It is the basic idea which is used to develop this immersed boundary method.

1. Computation scheme

1.1Governing equations

The governing equations for viscous flow of incompressible fluid can be written in the integral form as:

Continuity

whereΩis the integral domain and ?Ωis the boundary, dS, dΩare the elemental area vector and the elemental domain, respectively,Uis the velocity vector (u,v),ρis the fluid density,μis the fluid dynamic viscosity,pis the static pressure andFis the body force vector (fx,fy).

1.2Numerical scheme

1.2.1 Momentum prediction

The momentum equation can be solved for the cell center velocity vector with a given face flux and pressure. They can be either the initial values or intermediate values.

The Navier-Stokes equation is discretized in space with the collocated finite volume method and in time with the Euler implicit method. The linear equation set can be obtained as:

The VOS concept is used to consider the immersed boundary effect. The following formulations

are applied to calculate the body forces, whereu′′andv′′ represent the second correction values which can be linearly interpolated from the fluid and the immersed rigid body velocities as done by NG[13]:

whereαis the total solid fraction within a cell. The body force can be rewritten as follows

From the Newton’s third law, we can obtain the forces on the rigid body.

1.2.2 Pressure correction

The velocity does not always satisfy the continuity equation after the momentum prediction and it is necessary to correct it based on the mass conservation equation. A pressure correction equation can be derived as in literature[14].

The pressure can be corrected by solving the above equation. The face flux and the velocity can also be updated in this manner.

1.2.3 Numerical method of VOS

A link data structure is used for the immersed boundary containing a number of points. An efficient Lagrange point tracking algorithm on a fixed grid is applied to search the moving boundary grid points to improve the code performance[15]. The link direction is decided by the solid areas always on the left side of the boundary. In order to calculate the solid volume fraction in each cell, an array of pointer data structure is defined and a local polygon link is formed in the shadowed area. After the local link formation, the transformation from link to graph is completed. The solid volume fraction is calculated based on these graphs[14]. There are some special cases which require careful treatments, the details of which can be found in Ref.[16].

Fig.2 Sketch of immersed boundary fronts

After calculating the solid volume fraction of all cells which cross the immersed boundary, it is necessary to find the cells which are totally contained in the solid area. In this article, a method based on the concept of the advancing front is proposed to identify those cells quickly. As shown in Fig.2, one seeks the cell faces that are in the solid area but not cross the immersed boundary, which would form the front link, and then one advances the fronts until they disappear. The unstructured gird around an ellipse and the partially enlarged diagram are shown in Fig.2. The bold solid line represents the edge link, the thin line represents the immersed boundary and the dotted line represents the grid mesh meeting with the immersed boundary.

In order to deal with the multi-body case, the procedure is repeated for each body surface. As shown in Fig.3, there are two cylinders, so the VOS calculation procedure will be carried out twice.

Fig.3 Sketch of multi-body case

2. Presentation of results

2.1Lid driven cavity flow

In order to verify the present immersed boundary method, a benchmark problem is considered. This method is also used to simulate the flow past a stationary cylinder as in literature[17-20].

Fig.4 The sketch of the computational domain and grids

Fig.5 VOS distribution

Fig.7 Results forRe=400

Fig.6 Results forRe=100

Fig.8 Results forRe=1000

Fig.9 Computational domain

Fig.10 Contours of pressure around two cylinders moving to each other atRe=40 at three different stages. The first graph is obtained with a coarse grid and the second one is obtained with a refined grid

The lid driven cavity flow is a classic benchmark case, which is usually used to evaluate numerical methods and codes[21,22]The computational domain is shown in Fig.4, which is a 1.1 m×1.1 m cavity square. In order to test the present numerical method and codes, a boundary layer of 0.1m in width is assumed to be the solid zone on bottom and right sides. The VOS distribution is shown in Fig.5.

Fig.11 Contours of vorticity around two cylinders moving to each other atRe=40 at three different stages. The first graph is obtained with a coarse grid and the second one is obtained with a refined grid

Ghia’s results are a set of the most popular experimental data[23], which are extensively used, andhere they are regarded as the standard. In this article, simulations with different Reynolds numbers ranging from 100 to 1 000 are carried out and the results are shown in Figs.6 through 8. The Reynolds number is expressed as

whereUmaxis the lid moving velocity anddis the width of the cavity.

As shown in Fig.6, the velocity distribution agrees well with the experimental data. There are two small eddies and one dominant eddy. The two small eddies grow with the increase of Reynolds number and the center of the dominant eddy moves downwards. The difference of the velocity distribution between numerical results and experimental data increases with the increase of Reynolds number.

2.2Two cylinders moving to each other

This case was firstly used by Wang in a series articles for verification. Two cylinders move to each other with a velocity of 1 m/s and the diameter of both cylinders is 1 m as shown in Fig.9.

Two sets of meshes with 464 052 and 629 268 triangle cells with refinement near the cylinder motion zone are used to simulate the multi-body and fluid flow interaction. The contours of the pressure and the vorticity at three different instants are shown in Fig.10 and Fig.12, respectively, and the top one is the result obtained with a coarse grid and the bottom one is obtained with a refined grid. As shown in Fig.10 and Fig.11, the pressure and velocity contours are similar at different instants.

Fig.12 Contours of vorticity

The comparison of the vorticity att=24 with that in literature[2]as shown in Fig.12 shows that the present method is reasonable and accurate.

3. Conclusions

A numerical method on the fluid-structure interaction based on the immersed boundary method and the volume of solid method is extended to unstructured grids. This method can be applied to any existing pressure based finite volume CFD solver without complex procedures.

To verify the present method, the flows past a stationary and an oscillating cylinder are simulated and the results agree well with those in literature. The results for two cylinders moving to each other are reasonable and consistent with Wang’s data, which indicates that this method is accurate.

Acknowledgement

This work was supported by the Fundamental Research Fund of Harbin Engineering University (Grant No. HEUFT 09005).

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February 12, 2011, Revised April 7, 2011)

* Biography: MING Ping-jian (1980-), Male, Ph. D., Lecturer

2011,23(4):476-482

10.1016/S1001-6058(10)60138-2