Fluid Structure Interaction

We will interest now to the different interactions a fluid and a structure can have together with specific conditions.

Fluid structure model

To describe and solve our fluid-structure interaction problem, we need to define a model, which regroup structure model and fluid model parts.

We have then in one hand the fluid equations, and in the other hand the structure equations.

Figure 1 : FSI case example

The solution of this model are $(\mathcal{A}^t, \boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s)$.

ALE

Generally, the solid mechanic equations are expressed in a Lagrangian frame, and the fluid part in Eulerian frame. To define and take in account the fluid domain displacement, we use a technique name ALE ( Arbitrary Lagrangian Eulerian ). This allow the flow to follow the fluid-structure interface movements and also permit us to have a different deformation velocity than the fluid one.

Let denote $\Omega^{t_0}$ the calculation domain, and $\Omega^t$ the deformed domain at time $t$. As explain before, we want to conserve the Lagrangian and Eulerian characteristics of each part, and to do this, we introduce $\mathcal{A}^t$ the ALE map.

This map give us the position of $x$, a point in the deformed domain at time $t$ from the position of $x^*$ in the initial configuration $\Omega^*$.

Figure 2 : ALE map

$\mathcal{A}^t$ is a homeomorphism, i.e. a continuous and bijective application we can define as

$\begin{eqnarray*} \mathcal{A}^t : \Omega^* &\longrightarrow & \Omega^{t} \\ \mathbf{x}^* &\longmapsto & \mathbf{x} \left(\mathbf{x}^*,t \right) = \mathcal{A}^t \left(\mathbf{x}^*\right) \end{eqnarray*}$

We denote also $\forall \mathbf{x}^* \in \Omega^*$, the application :

$\begin{eqnarray*} \mathbf{x} : \left[t_0,t_f \right] &\longrightarrow & \Omega^t \\ t &\longmapsto & \mathbf{x} \left(\mathbf{x}^*,t \right) \end{eqnarray*}$

This ALE map can then be retrieve into the fluid-structure model.