1. Model of a Sloshing Tank

2. Background
Oil tankers and transport trucks are two examples where sloshing can occur within a tank.
If sloshing is too extreme, it can create a non-uniform distribution of weight within the tank.
This model is a demo of free surface flow modeling using the Moving Mesh user-interface available in COMSOL Multiphysics
The method used to move and deform the mesh in COMSOL Multiphysics is known as an Arbitray Lagrangian Eulerian (ALE) method
The equations solved are the Navier-Stokes equations in a moving reference frame defined by the moving mesh

3. Model Definition
The free surface condition (no flow cross the surface boundary) is formulated with the built in tangent and normal coordinate system boundary conditions for the moving mesh, and a neutral-stress boundary condition for Navier-Stokes equations at the top surface.
Surface tension effects are neglected in this example, but could be included if needed
The flow, initially at rest, is driven by an oscillating gravity vector. This is to mimic a periodic ’tank’ motion. This can be visualized using a deformation plot, where the entire frame is deformed according to the gravity vector.
The example is made in 2D but could be generalized to 3D – you would only need a computer with more RAM compared to the 2D case.

4. Model Definition The fluid is glycerol with:
h = 1.49 Ns/m2
r = 1270 kg/m3
The walls of the ’tank’ allow free slip.
The gravity vector is (g_x,g_y)T, with:
g_x = g*sin(phi_0*sin(2*pi*f*t)),
g_y = -g*cos(phi_0*sin(2*pi*f*t)),
g = 9.81 m/s2
phi_0 = pi/128
f = 1 s-1

5. Model Definition – Mesh Movie* * set in presentation mode to view
The mesh deformation is computed according to the Navier-Stokes equations with the applied gravity load.
Note that higher-order finite elements are used not only within the tank to represent the flow field but also to track the free fluid surface.

6. Results - Movie* * set in presentation mode to view
Results showing the vertical fluid velocity (y-velocty) in colors and the x-y velocity as arrows.
The internal viscous force is the only energy dissipation mechanism in this example, therefore the wave amplitude is increasing and becomes large.
After a while higher-order modes of oscillation become visible.

7. Results
The wave height at the right side wall