Fluid Flow Simulation in Directed Energy Deposition (DED)#
Mathematical Model:
\[\begin{split}
\begin{align*}
&\text{Navier-Stokes Equations:} \quad \rho \frac{\partial \mathbf{v}}{\partial t} + \rho(\mathbf{v} \cdot \nabla)\mathbf{v} = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f} \\
&\text{Heat Transfer:} \quad \rho c \frac{\partial T}{\partial t} + \rho(\mathbf{v} \cdot \nabla)T = k \nabla^2 T
\end{align*}
\end{split}\]
Weak Formulation:
\[\begin{split}
\begin{align*}
&\int_{\Omega} \left(\rho \frac{\partial \mathbf{v}}{\partial t} \cdot \boldsymbol{\psi} + \rho(\mathbf{v} \cdot \nabla)\mathbf{v} \cdot \boldsymbol{\psi} + \nabla p \cdot \boldsymbol{\psi} - \mu \nabla^2 \mathbf{v} \cdot \boldsymbol{\psi} - \mathbf{f} \cdot \boldsymbol{\psi}\right) \,d\Omega = 0 \\
&\int_{\Omega} \left(\rho c \frac{\partial T}{\partial t} \phi + \rho(\mathbf{v} \cdot \nabla)T \phi - k \nabla^2 T \phi\right) \,d\Omega = 0
\end{align*}
\end{split}\]