Bo-qian Yan,Shuangqiang Wang,Gui-yong Zhang,3,Chen Jiang,Qi-hang Xiao,Zhe Sun

1.State Key Laboratory of Structural Analysis for Industrial Equipment,School of Naval Architecture,Dalian University of Technology, Dalian 116023,China

2. Department of Fisheriesand Oceans, Bedford Institute of Oceanography, Dartmouth,Canada

3.Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240,China

4.Key Laboratory of Traffic Safety on Track of Ministry of Education,School of Traffic and Transportation Engineering, Central South University,Changsha 410076,China

5.Joint International Research Laboratory of Key Technology for Rail Traffic Safety,Central South University,Changsha 410076,China

6.National and Local Joint Engineering Research Center of Safety Technology for Rail Vehicle,Central South University,Changsha 410076,China

Abstract:To solve the problem of inaccurate boundary identification and to eliminate the spurious pressure oscillation in the previously developed immersed smoothed point interpolation method (IS-PIM),a new sharp-interface IS-PIM combining mass conservation algorithm,called Sharp-ISPIM-Mass,is proposed in this work.Based on the so called sharp-interface method,the technique of quadratic local velocity reconstruction has been developed by combining with the mass conservation algorithm,which enables the present method improve the accuracy of the velocity field and satisfy the mass conservation condition near the boundary field.So the proposed method would not encounter the problem of spurious mass flux.In addition,a new form of FSIforce evaluation considering pressure and viscous force to perform a whole function from the fluid domain to fictitious fluid domain is introduced,which makes the present method obtain more accurate results of FSI force than the original one.Through the numerical studies of a number of benchmark examples,the performance of the Sharp-ISPIM-Mass has been examined and illustrated.

Key words:Sharp-interface technique,immersed type method,massconservation algorithm,fluid-structureinteraction (FSI)

Introduction

Fluid-structure interaction (FSI)problems are frequently encountered in engineering areas,such as automobile industry, aerospace engineering,hydro and wind industries[1].Due to the complex interaction between fluid behavior and structure behavior,lots of numerical methods have been developed well recently[2-3].Arbitrary Lagrangian-Eulerian (ALE)methods[4-5]are useful to solve FSI problems by the process of mesh regeneration.But it is powerless to handle the mesh deformation phenomenon when it comes to complex dynamic systems[6].Furthermore,the process of mesh regeneration is so timeconsuming that it is hardly to be used in large engineering projects.A new type meshless method called smoothed particle hydrodynamics(SPH)method has been introduced to solve the FSI problem[7-8].Using this mesh free method,one can easily capture different complex boundaries and characterized by free surface boundaries,multiphase interfaces and so on.Liu et al.coupled dynamic solid boundary treatment (SBT)algorithm with SPH method to improve the accuracy of the solid boundary[9].Furthermore,Zhang et al.[10]introduced a high-accuracy SPH method for a large-deformation solid to improve the completeness of the approximation function.Immersed boundary method (IBM)which distributes the FSI force into three or four layers of fluid points near the interface between fluid and solid has been proposed by Peskin[11].This method can solve the mesh distortion problem by using fixed fluid meshes instead of mesh regeneration and can be high efficient comparing with the SPH method[12].IBM is so attractive due to the accuracy and efficiency that an increasing number of researchers have developed new numerical methods based on IBM,whose characteristic is that moving solid mesh past fixed fluid mesh.However,the fluid solver within IBM adopt uniform fluid grids which have limitations when resolving fluid domains with complex shapes[13]and furthermore,IBM has a major shortcoming that is the assumption of the fiber like immersed structure,which may carry mass instead of occupying volume in the fluid domain.Therefore,immersed smoothed finite element method (ISFEM)[6,13-14],immersed smoothed point interpolation method (IS-PIM)[1]and node-based partly smoothed point interpolation method (NPS-PIM)[15]are proposed by researchers.In the scheme of IS-PIM,a new mesh-free method called smoothed point interpolation method (S-PIM)[16]is used to solve solid domain.S-PIM can effectively conquer the overly-stiff behavior,a disadvantage seen in finite element method (FEM),and it possesses excellent computational efficiency and accuracy when it comes to element distortion[17-20].The fluid solver of IS-PIM adopts Galerkin process based on the semi-implicit characteristic-based split (CBS)scheme[21].CBS scheme allows triangular element (T3)for 2-D cases and tetrahedron element (T4)for 3-D cases to discretize fluid domain,which means that when the fluid domain is relatively complicated,the process of meshing would be more convenient and easier for using T3 or T4 than using uniform fluid mesh[22-23].

It isnotable that using immersed-type methods to solve FSI problems cannot precisely capture the interface between solid domain and fluid domain due to the mismatching between the fluid mesh and the solid mesh[24].In other words,the fluid domain can only recognize the outermost points within solid domain,rather than the points at the interface between fluid and solid,as the boundary,which can be called fluid-represented boundary.In order to solve this problem,numerous methods have emerged,such as the cutting mesh method[25],hybrid Cartesian/ immersed boundary method (HCIB)[26],ghost-cell IBM[27]and so on.The overall reviews of revised immersed-type methods can be found in the works of Luo[28]and Sotiropouloset al.[29].

Jiang et al.[24]proposed the Sharp-interface IS-FEM(Sharp-ISFEM)to solve the inaccurate boundary problem based on IS-FEM.In his work,the sharp-interface revision was accomplished by local fluid velocity reconstruction with unstructured fluid mesh.The principle of this local velocity reconstruction was based on geometrical revision which was aimed at ensuring that the velocity of the outermost fluid nodes within the interface was consistent with the shape of solid boundary.By the way, according to the work of Gilmanov and Sotiropoulos[26],this local revision had second-order accuracy.Sharp-ISFEM showed better results in terms of velocity field of fluid especially in the vicinity of the interface comparing with the original IS-FEM.However,it should be pointed out that the spurious pressure of fluid near the boundary was stimulated by the geometrical revision which would lead to unbalance of the mass flux of fluid across the interface[30-31].This unbalance of the mass flux was the result from that the man-made velocity revision ignoring the physical consistency.That is to say,although the geometrical revision of fluid velocity is essential,it would lead to the fluctuation of mass flux and the spurious pressure,which can absolutely cause the incorrect evaluation of the pressure value as well as the FSI force.In this paper,the mass conservation algorithm[32],which could settle the problem of the unbalanced mass flux across the boundary by the revision of some points’pressure,isadopted.

In addition,the original IS-PIM has been using fictitious fluid domain to calculate FSI force.The pressure of fictitious fluid domain,which is identical to the solid domain,is computed through interpolation between fluid domain and fictitious fluid domain.However,the viscous force of fictitious domain is computed via the gradient of velocity of the fictitious fluid field instead of fluid field[6,13].As Fig.1 shows,using fictitious domain to calculate the viscous force can only obtain the viscous force within the fictitious field as the blue region represents but ignoring the viscous force at the interface as the red line denotes.This ignoring can cause significant numerical errors when it comes to the solid with large stiffness and fluid with low Reynolds number.To take the viscous force at the interface into consideration,the pressure and viscous force should be calculated as a whole function using fluid domain,including white region and blue region as Fig.1 shows,thus the gradient of the velocity at the interface can be captured.Then the pressure and viscous force acting on the fictitious domain can be obtained by interpolation.

1.Numerical method

1.1 Brief review on IS-PIM

The Sharp-ISPIM-Mass is based on the immersed smoothed point interpolation method (IS-PIM).To begin with,the brief reviews of the original IS-PIM which can be divided into fluid solver,solid solver,imposing FSI velocity condition and imposing FSIforce condition are introduced.

Fig.1 (Color online)The fluid domain and fictitious domain in IS-PIM

1.1.2 Semi-implicit CBSmethod

In this work,linear triangular (T3)elements are used to discretize the fluid domain.Therefore,the velocity and pressure of fluid can be obtained under unstructured mesh through FEM process[22].In addition,it is easy to generate unstructured fluid mesh for complex fluid domain using T3 elements,thus the velocity and the pressure of the specific element can be written:

1.1.3 Governing equations for solid

The governing equations of solid can be written asfollows

1.1.4 The analysisof nonlinear solidsusing S-PIM

Fig.2 Edge-based smoothing domain[16]

Using explicit tine integration based on central difference algorithm,one can obtain the discretized motion equation of nonlinear solids in the following forms

1.1.5 Imposing FSI velocity condition

1.2 Imposing FSI force condition

1.2.1 The original formof FSI force evaluation

The original form of FSI force evaluation is proposed by Zhang et al.[13]to solve FSIproblem with large deformation solid.FSI force is obtained through fictitious fluid nodes based on Lagrange grids.

Following the spatial and time discretization processof the CBSscheme,one can easily achieve the fully discretized formulations of the FSI force.The nodal FSI force acting on the solid domain can be obtained based on the following equations,where subscript “ fc” represents fictitious fluid.

where

In the original method,FSI force acting on the solid domain is calculated through the velocity of fictitious fluid nodes,and pressure term is computed by interpolation from fluid domain to fictitious fluid domain based on Eq.(28).However, the viscous force term is calculated by the velocity gradient of fictitious fluid domain instead of fluid domain as Eq.(29)shows.The problem is that if only considering the velocity gradient of fictitious fluid domain,the velocity gradient at the interface would be neglect.Therefore,the viscous force at the interface will be ignored,although it may be a little part of total FSI force.To improve this deficiency,a new form of FSI force evaluation is proposed in next section.

1.2.2 A new FSI force evaluation

In this evaluation,FSI force is still calculated through fictitious fluid domain.However,instead of using the velocity gradient of fictitious fluid domain nodes, the viscous force acting on the fictitious fluid domain is obtained through the interpolation from real fluid domain to fictitious fluid domain.Therefore,considering that the pressure term is also calculated through the interpolation from real fluid domain to fictitious fluid domain,the viscous force and pressure force acting on the interface can be treated as a whole function.

The key point of this new type of FSI force evaluation is that firstly the viscous term and pressure term should be considered to perform a whole function acting on the real fluid domain,then the resultant force can be obtained by interpolation using shape function from real fluid nodes to fictitious fluid nodes.To better understand this method,the resultant force of an infinite small area of fluid field is described in Fig.3 for 2-D case[34].

The-x direction total force acting on an infinite small area of fluid field can be written

Fig.3 An infinite small area of fluid field

The FSIforce acting on the fictitious fluid can be obtained through interpolation by shape function from fluid domain to fictitiousfluid domain:

2.Sharp-ISPIM-Mass

In the scheme of the IS-PIM,to solve the inaccurate boundary problem,the local velocity revision called sharp-interface technique is imposed.The process is similar to Sharp-ISFEM[24].Then the mass conservation revision is considered to ensure no additional mass flux flowing acrossthe boundary.

2.1 Localvelocity reconstructionusing quadratic interpolation

In this section,local velocity reconstruction is employed to enhance the accuracy at the interface between the fluid and solid.As Fig.4 shows,the reason of the inaccurate boundary problem is that when imposing the FSI velocity condition from solid domain to fluid domain,there are no fluid points at the solid boundary,which means that the fluid domain can only recognize the red line as the interface between fluid and solid instead of the real boundary.Therefore,there would be large errors when solving the fluid field especially near the interface.The red linecan be called as fluid-represented interface.

Fig.4 (Color online)Illustration of fluid-represent interface and local velocity reconstruction

Hybrid Cartesian/immersed boundary method (HCIB) was proposed by Gilmanov and Sotiropoulos et al.[26].They reconstructed the first layer fluid nodes in exterior fluid domain to maintain the accuracy at the interface.Balaras et al.[35]also proposed a similar reconstruction method using line normal which has second-order accuracy.Jiang et al.[24]adopted a novel sharp-interface method called Sharp-ISFEM reconstructing the velocity field of fluid nodes near the boundary by linear interpolation scheme,so the velocity field near the interface has been more stable and smoother than the original IS-FEM.However,it should be noticed that the velocity of fluid nodes near the interface is subjected to nonlinear distribution.Thus,in this paper,quadratic interpolation is used to reconstruct the velocity field of the fluid domain near the boundary.The specific process of quadratic interpolation can be seen in Ref.[26].

As Fig.4 shows,the green square points which are the nearest nodes to interface in the real fluid region are tagged as immersed points(Ipoints)whose velocity will be reconstructed.B points are the projection points shown with red diamond,which are theprojection pointsof I pointson theinterface.Then,extend the normal of element boundaries from a B point passing through the corresponding I point,and the intersection of the normal and the fluid element edges is denoted as C point shown with black triangular point.The tagging process can be seen in Ref.[24].When tagging process is finished,this velocity reconstruction for I point is straightforward and efficient using quadratic interpolation.The specific process of interpolation isasfollows.

The velocity componentiv is assumed to vary in quadratic manner along the length of normal directions,soiv can be obtained

IS-PIM combining this local velocity revision can be called as Sharp-ISPIM.It should bepointed out that the revision of velocity field can calculate the values near the interface more accurate.However,it may also lead to spurious pressure oscillation caused by this non-physical reconstruction[36-37].In the next section,the mass conservation algorithm,which can ensurethe physical continuity,isintroduced.

2.2 Strong formmassconservation algorithm

Sharp-ISPIM can provide an excellent framework for solving FSIproblems.However,dueto the local velocity revision,the physical discontinuity is stimulated near the boundary[38].Most crucially,additional mass flux is introduced so that there is spurious pressure fluctuation near the boundary.This additional mass flux will lead to large numerical errors inevitably.Furthermore,as Fig.5 shows,it is evident that the fictitious fluid field would interact with the real fluid domain,which means there would be additional fluid flowing across the interface.Jungwoo Kim et al.[39]proposed an algorithm called mass source/sink method that can remedy this mass loss effects, but the effect of this method is related to the mesh quality.Udaykumar et al.[32]introduced another method called cut cell method to make sure the fluid domain satisfy non-penetrate condition along the normal direction at the interface.This method is relatively efficient and physical stable.In this section,Sharp-ISPIM absorbs this mass conservation algorithm (Sharp-ISPIM-MASS)to eliminate the spurious pressure oscillation.

Fig.5 (Color online)Illustration of mass flux exchanging

According to the previous works[31-32,37-38],the pressure gradient along normal direction at the interface satisfies

3.Numerical examples

3.1 2-D lid-driven cavity flow with a rubber wall

In this numerical example,the 2-D lid-driven cavity flow with a rubber wall problem has been studied in Ref.[24],and the results of 3-D case can been seen in Ref.[40].The different results of the two forms of FSI force evaluation mentioned in Section 2.5 will be discussed.In addition,the results of IS-PIM,Sharp-ISPIM, and Sharp-ISPIM-Mass are all presented as a comparison.The geometries of fluid cavity and solid wall are presented in Fig.6.

Fig.6 (Color online)Computational dimensions of the 2-D liddriven cavity flow with a rubber wall

Fig.7 (Color online)The deformation of the solid and themaximum vu

Table 1 The values of maximum displacement uv obtained using original IS-PIM with two different forms of FSI force evaluation without mass conservation algorithm

By comparing with the reference value[41-42]of the maximum displacementyu ,it can be clearly found that the original FSI force evaluation will lead to larger numerical errors than the present force evaluation method does.The reason is that in this example,Reynold number is so small that the viscous force would be the major contribution to the solid displacement to a great extent.However,the original FSI force is calculated using the velocity gradient of fictitious fluid domain,so that this evaluation would ignore the viscous force at the interface,which has been introduced in Section 2.5.In the present FSI force evaluation,the pressure and viscous force are considered to perform a whole function acting on the fictitious fluid domain, so that the viscous force at the interface can be captured.It is evident that the result of IS-PIM with the present FSI force evaluation is more closer to the result of the reference,so in the rest of the paper,the present FSI force evaluation is adopted.Table 2 shows the values of maximum displacementsyu using three different types of IS-PIM with the present form of FSI force evaluation.

Table 2Thevaluesof maximum displacement yu using ori- ginal IS-PIM, Sharp-ISPIM and Sharp-ISPIM-Mass

According to the Table 2,the maximum displacement using three different typesof IS-PIM are considered.Generally speaking,the results obtained by these three methods have little difference.Specifically,the results using original IS-PIM and Sharp-ISPIM are the same,which meansthe impact of velocity field near the interface between fluid and solid is inconsequential because both of the revised local velocity and the original velocity have little difference.In addition,it is obvious that the result of the Sharp-ISPIM-Mass is closer than that of the reference because this method can ensure there is no additional mass flux flowing across the interface so that the pressure in the part of the FSI force will be more accurate than theother two methods.

The solid domain will achieve steady state contours when >10 s t .The contours of the vertical displacement of the solid field in different physical times using Sharp-ISPIM-Mass are displayed in Fig. 8.When 1s t< , the left side of the solid begin to bulge and the right side start to sink simultaneously.As the flow developing,the force from the fluid to solid is increasing so that the degree of the deformation of the solid domain is greater.When 1s<7 s t<,the deformation of the solid varies so obvious.But when t >7 s ,the variance of the deformation is hardly detectable until>13s t when the solid reach the steady state.In addition,the contours shown in Fig. 8 are similar to the resultsof Refs.[41-42].

Fig.8 (Color online)Computational dimension of flow past a cylinder

3.2 Flow past a cylinder

Fig.9 (Color online)The distribution of yu of solid wall vary with time changing

Table 3 Summary of results of drag coefficient for =40Re

According to Table 3,thedC value varies from 1.50 to 1.62 based on other researches’results.dC computed by original IS-PIM is relative larger,because the fluid-represented-interface is not exactly conforming to the solid boundary as Chapter 3 mentioned.As Fig.10 shows,the fluid can only recognize the red solid line within the solid domain as the solid boundary,so the pressure is relative larger due to the contracted boundary.The result of the Sharp-ISPIM issmaller than other researchers’results,because it adopts local velocity revision technique near the boundary.Although the velocity field takes the solid boundary into consideration,this velocity revision is only a geometrical revision,and the physical continuity is not considered.Therefore,the FSI force calculated by Sharp-ISPIM is inaccurate which would lead to incorrect results.When it comes to Sharp-ISPIM-Mass,combining the local velocity revision with mass conservation algorithm,the physical continuity is guaranteed and thedC result of Sharp-ISPIM-Mass shows a conforming value to other researchers.To better demonstrate the difference of these three methods, the contours of the velocity field isshown in Fig.11.

According to Fig.11,it can be seen that the fluid field near the solid boundary using original IS-PIM,the left figure in Fig.11(c),is largely rough and even the fluid flows across the boundary.Using the Sharp-ISPIM,the middle figure in Fig.11(c),can eliminate streamline-penetrated phenomenon seen in original IS-PIM but the velocity gradient of the fluid near the boundary is wiped out.That is to say the accuracy near the boundary is abandoned to solve the streamline-penetrated problem,which definitely give rise to larger numerical errors.When the velocity gradient near the boundary is smaller,the FSI force would be smaller. A better result can be obtained by using Sharp-ISPIM-Mass as right side figure in Fig.11(c)shows.The fluid field near the boundary is smoother and the velocity gradient near the boundary is more precise than that using other methods.So the Sharp-ISPIM-Mass can capture the solid boundary,at thesame timeensuring the accuracy of the FSIforce.

Fig.10 (Color online) The solid boundary and numerical interface of the flow past cylinder using original IS-PIM

The length of vorticity of the fluid behind the cylinder is also computed using these three methods,which are original IS-PIM,Sharp-ISPIM and Sharp-ISPIM-Mass,as Fig.12 shows and the results are summarized in Table 4.The results of the original ISPIM and Sharp-ISPIM-Mass are comparable to other researchers’.However,the results using Sharp-ISPIM is relative larger than other researchers.Therefore,it is safe to say,although the local velocity revision could make the velocity field near the boundary smoother and can solve the streamlinepenetrated problem,the mass conservation algorithm must betaken into consideration.

Table 4Summary of the length of thevorticity for =40Re

Fig.11 (Color online)The contours of the velocity field near the boundary using original IS-PIM,Sharp-ISPIM and Sharp-ISPIM-Massfrom left to right.(a),(b)and (c)represents thegradual amplification of thevelocity contour near the boun dary between solid and fluid

3.3 Falling disk under gravity

Fig.12 (Color online)The pressure contour and the streamline of the fluid

Fig.13 Computational dimension of Falling disk under gravity

In these three cases,the solid disk is modeled by St.Venant elastic material.The disk will reach a steady state with a constant velocity after a period of time.The temporal histories of the disk vertical velocityvv and vertical displacementvu using original IS-PIM,Sharp-ISPIM,and Sharp-ISPIMMass are shown in Fig,15.The results are comparative to the results conducted by Glowinski et al.[49]who did the research on a rigid falling disk in the viscous fluid.In addition,according to the reference,the steady velocity can be obtained by an empirical formula with the following form[50]

Fig.14 (Color online)The unstructured meshesof the solid domain and fluid domain

Fig.15 (Color online)The time histories of sphere vertical velocity vv and vertical displacement vu using original IS-PIM,Sharp-ISPIM,and Sharp-ISPIM-Mass

According to the Figs.15(a)-15(c)represents the vertical velocity of the cases 1,2 and 3 respectively.The green line is the value of the reference.It is evident that the steady velocity obtained by Sharp-ISPIM-Mass is closer to the reference values comparing to the other two methods.The absolute values of the steady velocity using original IS-PIM and Sharp-ISPIM are larger than reference values.Figures 15(d)-15(f)denote the vertical displacement of the disk of the cases 1,2 and 3 respectively.It is obvious that at the same time,the vertical displacement solved using Sharp-ISPIM-Mass is smaller than those solved using original IS-PIM and Sharp-ISPIM which is consistent with the curve of the vertical velocity.

Tables 5-7 have respectively summarized the specific values of the vertical velocity and the relative errors compared to the empirical formula at steady state with respective cases1,2 and 3.

Table 5 The values of vertical velocity and errors using ori- ginal IS-PIM, Sharp-ISPIM and Sharp-ISPIM-Mass with case 1

Table 6Thevalues of vertical velocity and errorsusing ori- ginal IS-PIM, Sharp-ISPIM and Sharp-ISPIM-Mass with case 2

Table 7Thevalues of vertical velocity and errorsusing ori- ginal IS-PIM, Sharp-ISPIM and Sharp-ISPIM-Mass with case 3

It is obvious that the numerical errors would increase as Reynolds number decreases using original IS-PIM.The reason of that is that as the Reynolds number decrease,the fluid field near the boundary will play a big role in the numerical results.Because of the inaccurate boundary,numerical errors will be larger as the Reynolds number decrease.The results using Sharp-ISPIM keep a relative large error,but it can diminish the error in case 3 with high Reynolds number.As mentioned that the fluid field near the boundary will play a big role in the numerical results in case 3 with high Reynolds number,and the revised local velocity will largely ameliorate the fluid field,so the error is significantly reduced.However,when the Reynolds number is large,this positive effect is invisible.It is evident that Sharp-ISPIM-Mass has better outcomes than other two methods and the error is reduced so significantly.The velocity and pressure contours at different time using Sharp-ISPIM-Mass are shown in Figs.16 and 17.

According the Figs.16 and 17,the velocity contours and pressure contours are smooth and stable.The plots are also similar to other researchers’results[15]. So it is safe to say,Sharp-ISPIM-Mass is more accurate and feasible when solving FSIproblem with moving solid comparing to previous method.

Fig.16 (Color online)The vertical velocity development along with time changing

Fig.17 (Color online)The pressurecontoursalong with time changing

3.4 Vortex excited elastic beam behind a circular cylinder

In order to validate Sharp-ISPIM-Mass is useful for FSI problem with large deformable structure at unsteady state,a flexible beam attached on a fixed cylinder interacting with the unsteady viscous flow,as a benchmark FSI numerical example,isstudied[51].

Fig.18 Computational dimension of vortex excited elastic beam behind a circular cylinder

The top and bottom boundary condition adopts non-slip condition.At the right side of the channel,the pressure is set to be zeros.The left side of fluid channel is set to be flow inlet,and the velocity profile of the inlet satisfies:

In this FSIsystem, the fluid domain is discretized with 18 452 nodes with 36 340 irregular triangular elements and the solid domain is discretized with 763 nodes with 1 318 irregular triangular elements which are enough to get an accurate and convergent results.ES-PIM is adopted to solve the structure with large deformation.The contours of the fluid pressure field with streamlines at =t 13.50 s,13.54 s,13.58 s and 13.62 s are in a full period are plotted in Fig.19.It is evident that the plate behind the cylinder settles into a self-excited oscillation due to the periodical vorticities shed from the cylinder.The pressureat the front of the cylinder get its maximum value and the low-pressure region islocated behind the cylinder.

The -x direction velocity contours at =t 13.00 s,13.50 s,14.00 s and 14.50 s are also shown in Fig.20.The results are very familiar to those of other researchers[24,51-52].

The vertical displacementvu of beam endpoint A (0.6,0.2)is calculated and compared with the references summarized in Table 8.The vertical displacement of A varying with time courses is also displayed in Fig.21.Point A keeps a period oscillation when the time is full enough.The present method can provide comparable and consistent results with other numerical methods[51-52].

4.Conclusion

Fig.19 (Color online)The contours of pressure and streamline of the fluid at =t 13.50 s,13.54 s,13.58 s and 13.62 s

Fig.20 (Color online)The contoursof x-direction velocity plot of the fluid at =t 13.50 s,13.54 s,13.58 s and 13.62 s

Table 8 Thevertical displacement value yu comparing to other researchers’results

Fig.21 The vertical displacement of A vu varies with time course

In this paper,a new non-boundary conforming method revised using sharp-interface technique combining mass conservation algorithm called Sharp-ISPIM-Mass is introduced.The method is the development of IS-PIM which can be used to solve FSIproblem with large structure deformation robustly.This new method can not only effectively improve the velocity field near the boundary,but also satisfy mass conservation across the interface between fluid and solid.Thus it can ensure that there is no addition mass flux flowing across the interface.In addition,a new form of FSI force evaluation,which considers pressure and viscous force to perform a whole function from the fluid to fictitious fluid,has been introduced.This new type of FSIforce evaluation can perfectly improve the accuracy of viscous force acting on the fictitious fluid domain and lead to more accurate FSI results.So after making up the short board of inaccurate boundary identification and spurious pressure oscillation,the present method can make full use of the benefit of immersed scheme,which makes the Sharp-ISPIM-Mass a good candidate for the numerical simulation of FSI problems.Furthermore,simulating FSI problems with middle even high Reynold number is promising.Given that Sharp-ISPIM-Mass could solve the inaccurate boundary problem and ensure mass conservation across the interface,adding superior turbulence model to CBSmethod would be the following work.

Acknowledgements

This work was supported by the High-technology ship research project of Ministry of Industry and Information Technology of China (Grant No.2017-614),the Joint Found for Equipment Pre Research and China Shipbuilding Industry Corporation (Grant No.614B042802-28),the Fundamental Research Funds for the Central Universities(Grant No.DUT2017TB05),the China Postdoctoral Science Foundation (Grant No.2018M641693),the Liaoning Revitalization Talents Program (Grant No.XLYC1908027),the Science Foundation of Hunan Province (Grant No.2019JJ50790)and the computation support of the Supercomputing Center of Dalian University of Technology.