Application of Incompressible Smoothed Particle Hydrodynamics Method for 3D Fluid Solid Interaction Problem

FANFAN SUN, MINGYI TAN, JING TANG XING

Fluid-Structure Interaction Research Group University of Southampton

Building 28/1015, Highfield Campus, University Road, Southampton UK

ffs1g09@soton.ac.uk Abstract: A general method for fluid solid interaction problem simulations has been developed in 3D algorithm using incompressible smoothed particle hydrodynamics (SPH) method. The solid is assumed to be rigid so it can be considered as moving boundaries for fluid. Using repulsive force has been proved to be an efficient boundary treatment for incompressible SPH method before with 2D examples. The advantage of this boundary treatment will be more obvious in 3D simulations of fluid-structure interaction problems since that it requires the fewest particle numbers on the boundaries compared with other boundary treatments. The algorithm can be applied to fluid solid interaction problems with deformable solid by using elastic or plastic solid theories. In this paper, 3D dam-breaking is used as an example to demonstrate the performance of this method and aircraft ditching is simulated. Key-Words: Smoothed Particle Hydrodynamics; Incompressible flow; fluid-structure interaction; boundary treatment; aircraft ditching 1 Introduction Smoothed particle hydrodynamics (SPH) method is a fully Lagrangian mesh free method which has been largely developed in the past 30 years. It has been successfully applied to the simulation of fluid motions [1-11] and extended to fluid-structure interaction problems [12-19]. Most of them use hypo-elastic theory or finite element method for the solid to consider deformation. However, it is preferred to use particles to represent both solid and fluid to exploit the advantages of mesh free method to the full and also to keep the model as simple as possible. In this stage of our method, the solid is assumed to be rigid and only the coupled motion is taken into account for the interaction. Hence, the solid can be considered as moving boundaries for the fluid.

For incompressible flow, there are generally two ways to impose incompressibility: one is to run the simulations in the quasi-incompressible limit by assuming a small Mach number to ensure density fluctuations within 1% [1, 20], which is known as weakly compressible smoothed particle hydrodynamics (WSPH); the other one is called truly incompressible SPH (ISPH) in which incompressibility is enforced by setting the velocity divergence to zero while solving a Poisson’s

equation for pressure at every time step [10, 11]. As proposed in the previous paper [21], using constant density for fluid to ensure the incompressibility allows the repulsive force to be an efficient boundary treatment for ISPH. Using this boundary treatment can reduce the particle numbers largely on the boundaries, especially in 3D cases compared with ghost particle boundary treatment which is the traditional boundary treatment for ISPH.

The results obtained from dam breaking case study will be compared with experimental and 2D results. The motion of aircraft ditching will be shown though further investigation about this problem will be produced in the short future. 2 Governing Equations Governing equations for fluid solid interaction include two parts: one is for fluid and the other is for rigid body. There is no special treatment for the interface by using SPH method. 2.1 Governing equations for fluid Fluid flows are governed by the classical Navier-stokes equations including continuity equation and momentum conservative equation.

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01=⋅∇+ v

dtdρ

ρ (1)

τgv⋅∇++∇−=

ρρ11 p

dtd

(2)

where ρ is the density, v is the velocity, P is the pressure, g is the body force and τ is the viscous stress tensor. For incompressible fluids, the mass density takes a constant, so the equation (1) reduces to

0=⋅∇ v (3) Splitting the momentum equation into two parts, one is to consider the effect of body force and viscosity and this introduces an intermediate velocity

nnn

tτgvv⋅∇+=

−+

ρδ12/1

(4)

Another part is to consider the pressure effect based on the intermediate velocity and the new velocity can be obtained

12/11 1 +

++

⋅∇−=− n

nn

Pt ρδvv

(5)

Taking divergence of (5) and substitute into (3) so the pressure Poisson’s equation can be derived

2/11 ++ ⋅∇=∆ nn

tP v

δρ

(6)

Assuming a Newtonian fluid, the viscous stress tensor in the momentum conservative equation is related to the velocity by the following equation

∂

∂+

∂∂

==i

j

j

ijiij x

vxv

ττ µ (7)

Hence equation (2) can be written as

vgv 21∇++∇−=

ρµ

ρP

dtd

(8)

Using chain rule, the gradients on the right hand-side of equation (4) can be approximated as:

−

−=

∂∂

ab

jb

ja

ab

ib

ia

aj

i

rxx

rvv

xv

(9)

The suffix ji, indicates tensor index of value 1, 2 or 3 and ba, indicates different particles. 2.2 Governing equations for rigid-body motion

To describe the motion and position of the solid three different coordinate systems are used: global coordinate, moving coordinate and body-fixed coordinate systems. These three coordinate systems coincide with each other initially. The moving system will move with the mass center (translation) and the body-fixed system will move with the whole body (both translation and rotation). The motion of a rigid body is combined of the translational displacements of its mass center in three directions and a rotational motion about the mass center with three axes. The most common way to describe the angular motion in a three dimensional space is the use of Euler angles. As in each time step the rotation angle should be very small, to avoid any singularity the Carl Dan angle description is adopted [22].

Fig.1 Carl Dan angle

A new orientation of the body can be derived by

assuming the system to rotate around x-axis with angleα first, and then rotate around the new y-axis with angle β , then rotate around new z-axis with angleγ . Hence, the transformation matrix from the moving coordinate system to the new body-fixed coordinate system is

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−+−−−+

=

−×

−

−=

βαβαβγβαγαγβαγαγβγβαγαγβαγαγβ

αααα

ββ

ββγγγγ

cccssssccssssccsccscsscsscccc

A

cossin0sincos0

001

cos0sin010

sin0cos

1000cossin0sincos

(10) The vector description in a new coordinate system can be obtained by the following transformation

Arr =' (11) The translation is described in the global system and rotation is described in the body-fixed system. The equations of motion for the translation and rotation can be expressed as

( ) MII

frm

=⋅×+⋅

=

ωωω

(12)

where f and M denote vectors of the external force and moment. 3 SPH Formulations The SPH formulation is based on the theory of integral interplant that uses kernel function to approximate delta function. The system is represented by a set of particles which carry all the physical properties as the material. The basic idea of this method is to approximate a function A(r) as

( )∑ −=b

bab

bba hW

AmA ,)( rrr

ρ (13)

abab b

b

a

ab

a

WPP

mP ∇

+=

∇ ∑ 22

1ρρρ

(14)

Hence, the velocity of fluid particles can be obtained by applying SPH formula to equations (4) and (5).

( )

+

−∇⋅++

=

∑

++

b ab

baabaabb

na

na

ruuWrm

gt

uu

222

12/1

4ηρ

µδ

(15)

abb

nb

na

bna

na W

PPmtuu '2

1

2

12/11 ∑

+−=

++++

ρρδ

(16)

For solid, the external force and moment only come from the pressure of the fluid. The solid particles on the interface are treated as boundaries for the fluid. The pressure values of these solid particles are calculated in the solution of Poisson’s equation. The total external force and momentum can be obtained from the summation of the pressure force and momentum. Then the equation of motion can be solved and new position can be obtained. 4 Boundary Treatments 4.1 Free Surface No special treatment is added on the free surface. Only the pressure is enforced to be zero for those particles on the free surface. 4.2 Solid Boundary The solid walls are simulated by particles which prevent the inner particles from penetrating the boundary. They are assumed to be able to provide a repulsive force to the inner fluid particles so the fluid will be stopped going out of the boundary.

( ) 200

21

rr

rr

rr

DrfPP

−

= (17)

It is zero when r>r0 so that the force is purely repulsive. Mostly, 41 =p and , gHD 5= [1]. The length scale r0 is taken to be the initial spacing between the particles.

Since this boundary treatment is proved to be efficient, it is applied for 3D simulations to reduce the boundary particles as well as to simplify the model generation for the complex geometries. 5 Courant Numerical Condition Since this incompressible SPH method calculates pressure implicitly and other propertied explicitly, the size of time step must be controlled in order to have stable and accurate results. The following Courant condition must be satisfied [11]

max

1.0v

ht ≤∆ (18)

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ρµ /125.0

2ht ≤∆ (19)

where h is the initial particle spacing and maxv is the maximum particle velocity in the computation. The factor 0.1 ensures that the particle moves only a fraction (in this case 0.1) of the particle spacing per time step. Another constraint is based on the viscous terms [23]. 6 3D Simulations and Results 6.1 Dam-breaking Case Study Dam breaking problem is a classic benchmark problem for assessment of fluid simulations. The water column is 0.1m wide and 0.18 m high. The time step size is 45 −e s, particle spacing is 0.003m, quadratic kernel function is used, and smoothing length is chosen as 1.333 times of the spacing. Assuming all the particles are in the hydrostatic state at the initial stage. The motion of fluid at 0.005s and 0.007s are shown below in Fig. 2 and Fig. 3. The results obtained from two phase flow, 2D and 3D simulations are compared with the experimental data [24].

Fig. 2 fluid motion at 0.005s

Fig. 3 fluid motion at 0.007s The 3D algorithm can produce good simulation of fluid motions as shown in Fig. 2 and Fig. 3.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

experimental2dmulti-phase3d

non-dimensional time T=t(g/H)0.5

non-

dim

ensio

nal l

eadi

ng e

dgX

=x/H

Fig. 4 results from experiment, 2D, 3D and multi-phase flow simulations From Fig. 4, it is obvious that 3D simulation can provide better results compared with 2D and multi-phase simulations with quadratic kernel function. 6.2 Aircraft Ditching It is important for the manufacturer to consider about the ditching ability of the aircraft on water in case of emergency. The experiments are always very expensive so that numerical simulations become an attractive tool to investigate these problems. However, numerical method for large deformation fluid solid interaction problem is still in the developing stage. SPH as a mesh free method has the advantage of simulating large deformation problems. Therefore, it is practicable to consider applying SPH to aircraft ditching simulations. Both the solid and fluid are simulated using particles. The geometry of the aircraft is based on the existing experiment [25].

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Fig. 5 initial position of aircraft ditching

Fig. 6 motion of aircraft ditching The motion of the aircraft is reasonable in this stage. More data including velocity and attitude should be obtained in the short future. 7 Conclusion Incompressible SPH with repulsive force as boundary treatment has been applied to 3D fluid and fluid solid interaction simulations. The applicability to fluid rigid body interaction is proved with dam breaking case study. The motion of aircraft ditching is reasonable in this early stage. More data will be provided shortly. It can be foreseen that it will be practicable to be extended to deformable bodies by applying suitable theories on the solid. References: [1] Monaghan, J.J., Simulating free surface flows

with SPH. J. Comput. Phys., 1994. 110(2): pp. 399-406.

[2] Joseph P. Morris, P.J.F., Yi. Zhu, Modeling low Reynolds number incompressible flows using SPH. Journal of Computational Physics, 1997. 136: p. 214-226.

[3] Lee, E.-S., et al., Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. J. Comput. Phys., 2008. 227(18): p. 8417-8436.

[4] Becker, M. and M. Teschner, Weakly compressible SPH for free surface flows, in Proceedings of the 2007 ACM

SIGGRAPH/Eurographics symposium on Computer animation. 2007, Eurographics Association: San Diego, California.

[5] Dalrymple, R.A. and O. Knio. SPH Modeling of Water Waves. 2001. ASCE.

[6] Ellero, M., et al., Incompressible smoothed particle hydrodynamics. J. Comput. Phys., 2007. 226(2): p. 1731-1752.

[7] Ferrari, A., et al., A new 3D parallel SPH scheme for free surface flows. Computers & Fluids, 2009. 38(6): p. 1203-1217.

[8] Hosseini, S.M., M.T. Manzari, and S.K. Hannani, A fully explicit three-step SPH algorithm for simulation of non-newtonian fluid flow. International Journal for Numerical Methods for Heat & Fluid Flow, 2007. 17: p. 715-735.

[9] Monaghan, J.J., SPH compressible turbulence. Monthly Notices of the Royal Astronomical Society, 2002. 335(3): p. 843-852.

[10] Pozorski, J. and A. Wawrenczuk, SPH computation of incompressible viscous flows. Journal of Theoretical Applied Mechanics, 2002. 40: p. 917.

[11] Shao, S. and E.Y.M. Lo, Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources, 2003. 26(7): p. 787-800.

[12] Antoci, C., M. Gallati, and S. Sibilla, Numerical simulation of fluid-structure interaction by SPH. Comput. Struct., 2007. 85(11-14): p. 879-890.

[13] Farahani, M.H., N. Amanifard, and S.M. Hosseini, A fluid-structure interaction simulation by smoothed particle hydrodynamics. Engineering Letter 16, 2009.

[14] Johnson, G.R., R.A. Stryk, and S.R. Beissel, SPH for high velocity impact computations. Computer Methods in Applied Mechanics and Engineering, 1996b. 139(1-4): p. 347-373.

[15] Oger, G., et al., Two-dimensional SPH simulations of wedge water entries. Journal of Computational Physics, 2006. 213: p. 803-822.

[16] Rabczuk, T., S.P. Xiao, and M. Sauer, Coupling of meshfree methods with finite elements : basic concepts and test results. Communications in Numerical Methods in Engineering, 2006. 22(10): p. 1031-1065.

[17] Fourey, G., et al., Violent fluid-structure interaction simulations using a coupled SPH/FEM method. IOP Conf.Series: Materials Science and Engineering 2010. 10.

[18] Groenenboom, H.L.P., Hydrodynamics and fluid-structure interaction by coupled SPH-FE

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method. Journal of Hydroaulic Research 2010. 48: p. 61-73.

[19] Attaway, S.W., M.W. Heinstein, and J.W. Swegle, Coupling of smooth particle hydrodynamics with the finite element method. Nuclear Engineering and Design, 1994. 150: p. 199-205.

[20] Hu, X.Y. and N.A. Adams, A multi-phase SPH method for macroscopic and mesoscopic flows. Journal of Computational Physics, 2006. 213(2): p. 844-861.

[21] Sun, F., M. Tan, and T.J. Xing, Investigations of boundary treatments in incompressible smoothed particle hydrodynamics for fluid-structure interactions. Proceedings of the 2nd international conference on fluid mechanics and heat and mass transfer 2011, 2011: p. 92-97.

[22] Jia, S., Rigid Body Dynamics. 1987, Advanced Education Publisher, China.

[23] Cummins, S.J. and M. Rudman, An SPH projection method. J. Comput. Phys., 1999. 152(2): p. 584-607.

[24] Martin, J.C. and W.J. Moyce, Part IV. An Experimental Study of the Collapse of Liquid Columns on a Rigid Horizontal Plane. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1952b. 244(882): p. 312-324.

[25] McBride, E.E. and L.J. Fisher, Experimental investigation of the effect of rear-fuselage shape on ditching behaviour, in NACA Technical Note 2929. April, 1953: Washington.

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