Free Surface Flow (Navier-Stokes with Free Surface)

Mathematical Model

Consider the Navier-Stokes equations coupled with a level set or volume-of-fluid method for tracking a free surface:

\[\begin{split} \begin{align} \rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &= -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \\ \nabla \cdot \mathbf{v} &= 0 \\ \frac{\partial \phi}{\partial t} + \mathbf{v} \cdot \nabla \phi &= 0 \end{align} \end{split}\]

where \(\phi\) is the level set or volume fraction representing the free surface.

Boundary Conditions

• Inlet and outlet: Specify velocity or pressure conditions.

• No-slip conditions on solid boundaries.

• Implement the level set or volume fraction boundary conditions.

Initial Conditions

Specify initial conditions for velocity, pressure, and the level set or volume fraction.

Weak Formulation

Find \(\mathbf{v} \in H_0^1(\Omega)\), \(p \in L^2(\Omega)\), and \(\phi \in H^1(\Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega + \int_{\Omega} \frac{\partial \phi}{\partial t} q \, d\Omega + \int_{\Omega} \mathbf{v} \cdot \nabla \phi q \, d\Omega \\ &= 0 \end{align} \end{split}\]

for all test functions \(\mathbf{w} \in H_0^1(\Omega)\), \(q \in L^2(\Omega)\), and \(r \in H^1(\Omega)\).