Luhui Han, Xiangyu Hu

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SPH modeling of fluid-structure interaction*

Luhui Han, Xiangyu Hu

This work concerns numerical modeling of fluid-structure interaction (FSI) problems in a uniform smoothed particle hydrodynamics (SPH) framework. It combines a transport-velocity SPH scheme, advancing fluid motions, with a total Lagrangian SPH formulation dealing with the structure deformations. Since both fluid and solid governing equations are solved in SPH framework, while coupling becomes straightforward, the momentum conservation of the FSI system is satisfied strictly. A well-known FSI benchmark test case has been performed to validate the modeling and to demonstrate its potential.

Fluid-structure interaction (FSI), smoothed particle hydrodynamics (SPH), total Lagrangian formulation

Introduction

Fluid-structure interaction (FSI) can be found in many natural phenomena, such as birds flying and fish swimming. Meanwhile, it also plays a very important role in the design of many engineering systems, e.g., aircrafts, engines and bridges. Although the mechani- cal behaviors of the FSI systems are quite different, their common essentials are interactions between movable or deformable structures and internal or external fluid flows[1].

Since FSI problems usually involve nonlinear fluid and solid dynamics, which are too complex to be solved analytically, they mostly have to be analyzed by means of experiments or numerical simulations, and the individual maturity of computational fluid dynamics (CFD) and computational solid dynamics (CSD) in past decades enables it[2]. Accordingly, a number of individual CFD or CSD approaches have already been developed, such as the space-time finite-element method (FEM)[3]and arbitrary Lagrangian Eulerian formulation (ALE)[4]. Because all of these methods are mesh-based, i.e., require that the domain is discretized into individual mesh-elements, they have to take significant efforts on re-meshing to prevent the occurrence of severe mesh distortion[5]. In comparison to conventional mesh-based methods, particle-based meshless methods are intended to approximate the equations of continuum mechanics in the domain only by nodes (particles) without being connected by meshes[6]. Typical meshless methods, which have already been used successfully in solving FSI, are coupled models like smoothed particle hydro- dynamics and discrete element method (SPH-DEM)[6]and lattice Boltzmann method-discrete element me- thod (LBM-DEM)[7]. A common advantage of all these meshless methods is that the identification of moving interfaces and deformable boundaries can be handled straightforwardly[8]. Nevertheless, these coup- led methods for FSI are still limited for more general applications. They both chose partitioned coupling in the form of “CFD-CSD” by using different discretiza- tion schemes to simulate separately the behaviors of fluid and structure, and usually require elaborate consideration for the momentum conservation at fluid-structure interfaces.

SPH method was first developed for simulating astrophysics problems by Lucy[9]and by Gingold and Monaghan[10], and has recently been adapted to many relevant engineering problems, including heat and mass transfer, molecular dynamics, fluid and solid mechanics. Owing to both its significant advantages of handling large deformations in a purely Lagrangian frame in simulation of solid dynamics and its great convenience of capturing breaking, merging, and splashing features in simulation of free surface flows, Antoci et al.[11]successfully proposed an FSI mode- ling within a uniform SPH framework. Since the same type discretization methods are allowed for a common description of both the fluid and the solid dynamics in terms of pressure and velocity, both the kinematic and dynamic interface conditions at fluid and solid inter- faces became straightforward and easy to be imple- mented in conservative formulations. Like in Gray et al.’s work[12], the constitutive model, i.e. the linear elastic relation of Hooke’s law between stress and deformation tensors, was applied with the incremental Jaumann formulation. As pointed out in Ref.[11], this relation is rate-independent, incrementally linear and reversible. If it is integrated in time with a sufficiently small time step, it can be adopted as a constitutive model when small finite deformations are considered. Other applications by using the same constitutive formulation in solving solid dynamics problems can be found in Refs. [13,14]. However, observing their simulation results in Refs. [11, 14], one can clearly see that the particle distributions have been shifted after a period of simulation. The initial uniformly distributed particles finally formed into another pattern even in areas without stresses. That is because the conventional SPH method has a shortcoming of inconsistency and the fore-mentioned incremental constitutive model is just an approximation of Hooke’s law. To overcome this limit, Vignjevic et al.[15]proposed a total Lagrangian framework for simulating solid dynamics, where Lagrangian kernels were employed directly to solve momentum equation with respect to the reference configuration. Besides, this total Lagrangian formalism takes another ad- vantage of not suffering from tensile instability problems[16].

Inherited from Antoci et al.’s work[11], we pro- posed a new numerical modeling for FSI problems in this work, where fluid dynamics equations are dis- cretized with conventional Eulerian kernels in current configuration while solid governing equations are solved with Lagrangian kernels in the reference configuration. Finally, a well-known FSI benchmark test case has been performed to validate the current SPH modeling and to demonstrate its potential.

1. Governing equations

1.1 Fluid equations

The governing equations for the motion of an isothermal, Newtonian fluid in a Lagrangian frame of current configuration are the continuity equation

and the momentum-conservation equation

Based on the weakly compressible SPH ap- proach[11, 17], which is often used to simulate incom- pressible flows, a linearized equation of state is introduced to estimate the pressure from the density field via

1.2 Structure equations

In this work, the solid model of structure is con- sidered to be elastic and compressible. The governing equations for the motion of a structure have the form ofbalance laws including the balance of mass and momentum.

From a total Lagrangian point of view, the ba- lance of mass adopts the algebraic form given by

and

In particular, for isotropic materials under Hooke?s law, Eq. (11) simplifies to

2. Numerical modeling

2.1 SPH discretization for fluid equations

(1)Density evolution equation

(2) Momentum equation

Using the inter-particle-averaged shear viscosity

Based on the analysis of Turek et al.[20], the boundary condition on the fluid-structure interface can be achieved by exerting a no-slip condition for the flow with moving boundaries. Following the imple- mentation of Adami et al.’s wall boundary condition[21]and assuming the dummy particles in structure region used to mimic the interaction on the fluid-structure interface just coincide with the real structure particles, for each interacting particle pair, we can simply approximate the imagining pressureand velocityon dummy particlewith

and

Alternatively, one can also choose the same extra- polation scheme proposed in Ref. [21] to evaluate them.

and

Since the negative pressure occurs in the wake in the FSI benchmark cases, which usually leads to particle clumping and void regions during the simulation, one also needs a remedy to solve this tensile instability problem in fluid regions. A number of solutions have already been proposed and validated, such as in Refs. [16, 17]. In this work, we choose transport-velocity scheme exactly the same as that in Ref. [17] (see details therein).

2.2 SPH dicretization for structure equations

where

is the gradient of a Lagrangian kernel function. Note that Eqs. (28) and (29) are only calculated once at the beginning as it is only related to the initial or reference configuration.

(1) Density evolution equation:

(2) Momentum balance equation:

with the 1st Piola-Kirchhoff stress tensor of particlebeing

and Green-Lagrange strain tensor

Finally, according to the Newton’s third law of motion, each structure particle involved in Eqs. (26) and (27) takes an equal and opposite reaction from the interacting fluid particle, therefore, the forces per unit mass owing to fluid-structure interactions can be given easily by

and

Notice that, the gradient of elastic stress in struc- tures is calculated in the reference configuration within the total Lagrangian formulation, however, the fluid-structure interaction forces must be obtained in the current configuration.

3. Numerical test

In this section, we present the validation results for the well-known 2-D benchmark FSI problem, defined by Turek and Hron[20], as shown in the Fig. 1, where the flow passes a fixed circular cylinder with a flexible beam attached to its downstream side. This test model has been frequently used as a large- displacement benchmark validation case for 2-D FSI solvers[5, 23].

Fig. 1 Model setup for flow-induced vibration of a flexible beam attached to a rigid cylinder[5]

Concerning boundary conditions, no-slip walls are exerted on the top and bottom sides of the domain while outflow condition on the right side. Fluid flows into the domain from the left side with a parabolic velocity profile

where denotes the end time of the starting proce- dure.

Fig. 3 Trajectory of Point A

Table1Comparison results for FSI2 test case

Fig. 4 (Color online) The fluid velocity field and beam defor- mation at different time instances marked in Fig. 2. The top panel of each subfigure shows the deformation of the beam with solid particles colored by contours of von Mises stress. The bottom panel presents the distribution of axial velocity component of the fluid

In Fig.4 we show the beam deformation at four different time instances in a typical oscillation cycle. Note that, since there is no elastic stress at the free end of the beam,particles in that region keep regular distribution all the time. This is not able to achieve by the previous SPH formulations based the incremental Jaumann formulation which uses Eulerian kernel in current configuration.

4. Conclusion

In this paper, we have proposed a numerical modeling approach for simulating FSI problems in a SPH framework, where the fluid governing equations are discretized with conventional Eulerian kernel in current configuration and the solid governing equa-tions with Lagrangian kernel in the reference con-figuration. To avoid tensile instability in fluid regions, a transport-velocity technique is employed to remedy the distribution of fluid particle. By using a total Lagrangian SPH formulation dealing with the structure deformations, we also apply a correction matrix to restore first order consistency and rotational invariance. It successfully avoids the occurrence of artificial strain and stress when a rigid coordinate transformation occurs on structures. Since both fluid and solid governing equations are discretized with SPH formulation, coupling becomes straightforward and meanwhile the momentum of an FSI system is strictly conserved. In order to validate the modeling and demonstrate its potential, a typical FSI benchmark test case is carried out.

Acknowledgement

The authors gratefully acknowledge the financial support by Deutsche Forschungsgemeinschaft (Grant No. DFG HU1527/6-1) for the present work.

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(October 22, 2017, Accepted December 7, 2017)

?China Ship Scientific Research Center 2018

Luhui Han (1985-), Male, Ph. D. Candidate,

E-mail: Luhui.han@tum.de

Xiangyu Hu,

E-mail: Xiangyu.hu@tum.de