Numerical simulation of fluid–structure interaction by SPH
Introduction
In many engineering applications, the forces exerted by a fluid flow on the confining solid boundaries do not modify significantly the geometry of the boundaries. In this cases, the fluid flow can be studied as occurring within rigid boundaries, and the forces applied on the solid boundaries can be obtained after the characteristics of the fluid motion have been determined.
On the other hand, whenever the characteristic times of the motion of the fluid flow and of the solid boundaries are comparable, it is necessary to couple the dynamics of the two media. These fluid–structure interaction (FSI) problems can be solved by employing either a simultaneous (or direct) solution or a partitioned (or iterative) solution. A description of the two procedures can be found in [1], together with the explanation of their main advantages and drawbacks. The simultaneous technique is particularly convenient when the interaction between the structure and the fluid is very strong (and the displacements of the structure are important). Structures are usually described by Lagrangian formulations, whereas fluids are often described by Eulerian formulations. The coupling of the two media is usually obtained by an Arbitrary-Lagrangian–Eulerian (ALE) formulation for the fluid. Significant contributions [1], [2], [3] have been proposed in the simulation of FSI problems in this context. Rugonyi and Bathe [1] perform a simplified stability analysis of the interface equations and study the long-term dynamic stability of FSI systems by use of Lyapunov characteristic exponents. They also show the solution of some FSI problems, as the dynamics of spring-loaded valves in fuel pumps, that indicate the actual possibility to simulate complex coupled phenomena. Recent developments in the simulation of viscous incompressible and compressible fluid flows with structural interactions are discussed in [2]. Le Tallec and Mouro [3] simulate the dynamics of an hydroelastic shock absorber adopting an ALE formulation for the fluid equations.
An alternative approach to the numerical simulation of FSI problems consists in the description of both the fluid and the structure motion by a Lagrangian formulation. This can be especially effective when studying problems characterized by large displacements of the fluid–structure interface and by a rapidly moving fluid free-surface. An example of these problems is the FSI inside safety valves for pressure reduction, where an elastic plate deforms owing to water pressure, allowing part of the fluid to flow out at atmospheric conditions, thus causing a pressure relief in the connected pipe. In this kind of problems, the use of Lagrangian techniques for both the solid and the fluid part of the problem appears promising, as it permits to easily follow in time the motion of the fluid–solid interface and to simulate the free-surface of the fluid without any specific treatment. In particular, encouraging results have been recently obtained by the smoothed particle hydrodynamics (SPH) technique (see [4] for a recent review of the method), which allows to obtain numerical solutions of the continuum equations by defining the variables at a set of suitable moving points, reconstructing the continuous field by means of interpolation functions centred on each moving point.
The SPH technique was first developed in astrophysics by Lucy [5] and by Gingold and Monaghan [6]. It was then successfully applied to the study of various fluid dynamics problems, such as free-surface incompressible flows [7], and viscous flows [8], [9]. Since the early 1990s, SPH was applied also to the simulation of elasticity and fragmentation in solids: in particular, Libersky et al. [10] modelled the elastic response of solid structures by an incremental formulation of Hooke’s law.
SPH has been also used to simulate the interaction between different fluids [11], [12], different solids [13] and between fluids and structures [14] in presence of explosions. In some commercial codes, an SPH description of the fluid motion is coupled to a finite element formulation for the solid dynamics, in order to simulate FSI problems.
The present paper discusses a FSI model where both the fluid and the solid parts are modelled by SPH. Aim of the model is the analysis of FSI problems where large elastic displacements of the solid occur, while rapidly moving free-surfaces characterize the fluid motion.
The reliability of the numerical results yielded by the proposed SPH FSI model is checked against laboratory data obtained during a simple 2D interaction experiment.
Section snippets
Equations of motion
The motion of a continuum subjected to the action of gravity, in isothermal conditions, is described by the continuity equationand by the momentum equationwhere t is time, ρ is density, vi is the velocity vector, xi is the position vector, gi is the gravity vector, σij is the stress tensor and the notation implies summation over repeated indices.
The stress tensor can be decomposed into its isotropic and deviatoric parts:where is pressure,
Fluid–structure interaction model
Two different sets of particles are used for solid and fluid. All the particles located farther than 2h from the interface interact, of course, only with particles of the same species (particle a2 in Fig. 6). On the other hand, when SPH interpolation is performed on particles closer to the interface (like particle a1 in Fig. 6), particles of both media are involved. The simplest approach in this case consists in extending the summations in (10), (11) to all the particles b regardless of their
Deformation of an elastic plate subjected to time-dependent water pressure
The proposed SPH interaction model has been validated by comparison of the numerical results with data measured during suitable laboratory experiments. In these experiments an elastic gate, clamped at one end and free at the other one, interacts with a mass of water initially confined in a free-surface tank behind the gate.
Conclusions
A fluid–structure interaction model based on the SPH method has been described. In the model, both the fluid and the solid phases are discretized by SPH particles. Once the fluid–solid interface has been defined, coupling conditions are imposed to particles close to it. In particular, the action of the fluid on the solid is computed through the evaluation of an approximated surface integral of fluid pressure. On the contrary, the action on the fluid is computed by linear spatial interpolation
Acknowledgements
We are also grateful to Roberto Allieri and Ivano Brivio for their valuable help in the execution of the experiments.
We acknowledge Dresser Italia S.r.l. for the financial support to the present research.
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