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Fluid-Structure Interaction

1.1. Introduction

Recently, several new problems have been formulated in the area of fluid-structure coupling, for example in the automotive industry with the dynamics of airbag inflation and fluid sloshing inside tanks; in aeronautics with the fluttering phenomenon affecting airplane wings, which involves a coupling between the vibrational dynamics of a structure and the flow of a fluid; and in the transportation industry with studies on noise reduction inside vehicles based on vibroacoustic analysis.

Each and every structure in contact with a fluid is subject to phenomena involving mechanical fluid-structure couplings to some extent. This kind of multiphysics coupling often significantly affects the dynamic behavior of mechanical systems. Taking it into account is one of the major challenges in calculating the dimensions of structures, especially when the objective is to ensure that their design meets the necessary safety requirements.

In this chapter, we will examine problems relating to the interaction of a structure with fluids both at rest and in flow. We will give a description of the motion of the fluid based on vibration theory, considering small vibrations in the structure and fluctuations in the pressure of the fluid around a stable equilibrium state, and we will present the relevant equations in the case of flowing fluids and the corresponding numerical methods for calculating couplings with dynamic structures.

1.2. Fluid-structure interaction problem

The mechanical coupling between the two media acts in both directions at their surface of contact: deformations in the structure resulting from the forces applied by the fluid flow modify the state of the fluid-structure interface; this affects the flow conditions of the fluid, which induces a change in the forces exerted on the structure at the interface, thus bringing the interaction cycle to a close.

Figure 1.1. Fluid-structure coupling mechanism. For a color version of this figure, see www.iste.co.uk/elhami/interactions.zip

The fluid-structure interaction is described as the exchange of mechanical energy between a fluid and structure. This definition encompasses a wide range of problems. We can classify these problems using two criteria according to the physics of the problem at hand. The first criterion, proposed by Axisa [AXI 01], is based on the nature of the fluid flow. If the flow is negligible or non-existent, we say that the fluid is stagnant. Otherwise, we say that the fluid is flowing. In the first case, the objective is to describe small movements of the fluid and the structure around an equilibrium rest state. In these conditions, we choose to describe the dynamics of the interaction as a function of frequency; the equations describing the behavior of the structure and the fluid are written in terms of the reference (rest) state and generally lead to linear problems. In the second case, the objective is to establish a description of larger scale motion in the fluid and/or the structure. In these conditions, we choose to describe the dynamics of the interaction as a function of time; the equations describing the behavior of the structure and the fluid are written in terms of the current state of the system and generally lead to nonlinear problems.

The second criterion considers the coupling strength, which may be defined as the magnitude of the interactions or exchanges between the two media.

A coupling is said to be strong if there are high levels of exchange between the two media, i.e. the fluid has a significant impact on the structure, and vice versa. A coupling is said to be weak if the effect of one of the media dominates that of the other (Figure 1.2).

Figure 1.2. Examples of fluid-structure interaction problems [GAU 11]. For a color version of this figure, see www.iste.co.uk/elhami/interactions.zip

Three dimensionless numbers have been suggested to classify these problems [DEL 01]:

• - The mass number MA is defined as the ratio between the density of the fluid ?f and that of the structure ?s: [1.1]

  This describes the significance of the inertial effects of the fluid and the structure. If its value is close to one, the inertial effects of the fluid are comparable to those of the structure, and so must be taken into account.

• - The Cauchy number Cy is the ratio between the dynamic pressure and the elasticity of the structure, which is quantified by Young's modulus E. [1.2]

  This indicates the significance of the deformations induced by the flow. If this number is small, i.e. if the structure is rigid or the fluid velocity is small, structural deformations are negligible.

• - The reduced velocity Vr is the ratio between the characteristic flow velocity and the velocity of wave propagation inside the structure: [1.3]

If this number is large, the fluid dominates the problem from the perspective of time, and the dynamics of the structure are not important. By contrast, the dynamics of the structure increasingly dominate as this number tends to zero. If the number is close to 1, both dynamics carry similar weight in the problem.

These numbers are highly convenient for checking the importance of each phenomenon within the context of a given problem. However, as is the case for most dimensionless numbers, it is still difficult to define a priori threshold values applicable to all problems. In each problem, the large or small terms in the above will correspond to very different numerical values.

Using numerical simulations allows us to understand and predict the dynamic behavior of structures coupled with fluids, which is valuable in a number of industrial sectors. The numerical methods that we will use require us to solve the mathematical equations that model the behavior of the coupled fluid-structure system.

In general, the formulation of a coupled problem is based on the following description:

• - the structure problem is formulated in terms of the displacement; the goal is to describe the behavior of the structure as a function of the displacement u, strain e(u), stress s(u), and to solve the equations of this dynamic to find the u, e(u) and s(u) fields in the structure domain;
• - the fluid problem is formulated in terms of the pressure/velocity; the goal is to describe the behavior of the fluid as a function of the pressure p and the velocity v, to solve the equations of conservation of mass and momentum and to find the p and v fields in the fluid domain;
• - at the fluid-structure interface, the mechanical exchanges are represented, on the one hand, by considering the force f exerted by the fluid as a boundary condition for the structure problem and, on the other and, by considering the velocity imposed by the structure as a boundary condition for the fluid problem.

The energy exchanges between the fluid and the structure occur simultaneously. This needs to be taken into account by the numerical simulation. Coupled simulations can implement a single computational program to simultaneously solve the equations of the fluid and structure problems or alternatively can have two separate programs, one dedicated to the fluid problem and the other to the structure problem. The degree of complexity of the numerical simulation depends on the problem and methods of spatial and temporal discretization used to solve the equations of the problem.

Figure 1.3 proposes an overview of the most suitable general methods for simulating fluid-structure interaction problems:

Figure 1.3. General methods for numerically simulating fluid-structure interactions

1.2.1. Fluid-structure coupling methods

There are several suitable coupling methods for the kinds of problems that we typically encounter. The following methods are used for stagnant fluids:

• - The decoupled method finds the load or hydrostatic pressure on the structure, and then uses the results as an input to solve the deformation in the structure problem.
• - Acoustic fluid formulations (in terms of frequency) allow small displacements around the equilibrium position of a structure to be determined. If the fluid is heavy, the vibrations of the structure and the fluid are strongly coupled. This coupling is reflected in the distinct natural frequencies of the modes, and the shapes of these modes. These methods use formulations that can be either non-symmetric (u, p) or symmetric (u, p, f). They were proposed by Morand and Ohayon [MOR 95] and illustrated by Sigrist [SIG 11].

Frequency-based formulations of the fluid potential are applicable to problems with stagnant fluids, but can also be used to describe the elevation of a free surface subject to sloshing. The goal is to determine the motion of the free surface in order to find the pressure variations along the walls. These methods are similar to acoustic fluid formulations, which use either symmetric or non-symmetric expressions for the coupling equations, written as (u, p0) and (u,...