# Stability Evaluation of a Simplified Reclaimer Using Fluid-Structure Interaction
## 2 Materials and Methods
### Mathematical Model
#### 2.1.2 Numerical Model  
The FSI numerical model comprises the CFD and MDB models, which are interconnected. While the fluid forces act as external loads on the structure, the displacement and velocity of the structural interface characterize the boundary conditions for the CFD problem [26]. In the CFD analysis, the flow of the fluid follows the conservation equations of mass, momentum, and energy, and the transport equations for the turbulent variables are applied. The partial derivatives are numerically approximated and applied to several small control volumes. Boundary conditions are applied and the solution of the fluid properties is found after several computational iterations [27]. The mass and momentum equations are given by Equations (3)-(6):  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho u)+\frac{\partial}{\partial x}(\rho uu)+ \frac{\partial}{\partial y}(\rho uv)+\frac{\partial}{\partial z}(\rho uv)=- \frac{\partial P}{\partial x}+\mu\left(\frac{\partial^{2}u}{\partial x^{2}} +\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}} \right)+\frac{\partial}{\partial x}\left(\rho k\frac{\partial k}{\partial x }\right)-\rho\varepsilon+S_{Mx} \tag{4}\]  
\[\frac{\partial}{\partial t}(\rho v)+\frac{\partial}{\partial x}(\rho vu)+ \frac{\partial}{\partial y}(\rho vv)+\frac{\partial}{\partial z}(\rho vw)=- \frac{\partial P}{\partial y}+\mu\left(\frac{\partial^{2}v}{\partial x^{2}} +\frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}} \right)+\frac{\partial}{\partial y}\left(\rho k\frac{\partial k}{\partial y} \right)-\rho\varepsilon+S_{My} \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho w)+\frac{\partial}{\partial x}(\rho wu)+ \frac{\partial}{\partial y}(\rho wv)+\frac{\partial}{\partial z}(\rho wx)=- \frac{\partial P}{\partial z}+\mu\left(\frac{\partial^{2}w}{\partial x^{2}} +\frac{\partial^{2}w}{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}} \right)+\frac{\partial}{\partial z}\left(\rho k\frac{\partial k}{\partial z} \right)-\rho\varepsilon+S_{Mz} \tag{6}\]  
where \(\rho\) is the specific mass of the air, \(x\), \(y\), and \(z\) are the spatial directions, \(u\) is the velocity in the \(x\) direction, \(v\) is the velocity in the \(y\) direction, \(w\) is the velocity in the \(z\) direction, \(P\) is the pressure, \(\mu\) is the air dynamic viscosity, \(k\) is the turbulent kinetic energy, \(\varepsilon\) is the dissipation of the turbulent kinetic energy, and \(S_{Mx},\ S_{My},S_{Mz}\) are general source terms (these terms can represent any other external forces acting on the fluid that are not accounted for by the standard terms in the Navier-Stokes equations. For example, in magneto hydrodynamics (MHD), these terms might represent electromagnetic forces). The algorithm used to overcome the pressure-velocity coupling was PIMPLE. The \"PIMPLE\" algorithm in OpenFOAM stands for \"Pressure-Implicit with Multiple Picard-Loops\". It is an extension of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm and is used for solving the pressure-velocity coupling problem in computational fluid dynamics (CFD) simulations. PIMPLE is designed to handle transient and steady-state incompressible flows, particularly those with unsteady or turbulent behavior. The temporal scheme applied was the forward Euler scheme, a first-order scheme. The spatial order was Gauss linear and Gauss upwind, both first-order schemes. Multi-body dynamics is a subject that deals with the dynamic behavior of mechanical systems consisting of a series of bodies connected by kinematic constraints imposing restrictions on their relative motion [28]. The analysis is based mainly on the Lagrange equations, based on the energy conservation laws. The variables to be solved are the mass centers of each body and their mass and inertia moments.