MOF test case for two fluids with sheared velocity field

This test case simulates two fluids in a disc box.

The verification criteria is:

• The main volume fraction;
• The symmetric difference area.

Configurations

Physical domain and geometry

The simulations take place in a box of fluid of coordinates (0,0) and (1,1). The second fluid is placed in a disk positioned at (0.5,0.75).

Fluids' properties

Both of the fluids considered have the same properties :

where \(\rho\) is the density, \(\mu\) the dynamic viscosity coefficient, \(\sigma\) the conductivity, C the specific heat capacity, \(\alpha\) the thermal expansion coefficient and \(T_{0}\) the reference temperature.

Note

Since we only have to solve the Phase-advection equation, the fluid properties have no great influence on our simulation.

Velocity field

The velocity field is in the sheared case . The velocity is given as: \(u(x,y)=(\alpha (\sin(\pi x))^2 \sin(\pi y) \cos(\pi y) \cos( \dfrac{\pi t}{\tau} ),\alpha (\sin(\pi y))^2 \sin(\pi x) \cos(\pi x) \cos( \dfrac{\pi t}{\tau} ) )\) where \(\alpha \) and \(\tau \) are respectively the amplitude and the period.

Phase advection

MOF is the numerical method used for Phase advection.

Runtime parameters

Spatial discretization

The variable CellsPerDirection is included to change the regular mesh. The default value of this variable is 32. For each direction of the grid, we consider \(\dfrac{L}{\Delta x}=CellsPerDirection\).

Time discretization

The time step is fixed for the numerical resolution but depends on the variable CellsPerDirection. Time step is defined as: \(\Delta t=CFL \dfrac{\Delta x}{V_{max}}\) where \(V_{max}= \dfrac{\alpha}{\sqrt2}\).

UseAnalyticReconstruction choice

The UseAnalyticReconstruction variable is used to choose between the analytic reconstruction and minimization. The tolerance angle and derivative are respectively set to \(10^{-5}\) and 0.

UseFilaments choice

The UseFilaments variable permits using Filaments at MOF method.

Comments

• The use of filaments is necessary for a large mesh, otherwise one finds inconvenient results. By refining the mesh, the numerical results without and with filaments become similar.
• The mesh fining also gives better numerical results.
• The results given with analytic reconstruction are not different from those given with minimization.

In the following section, both filaments and analytic reconstruction are activated.

Results

Numerical results

Figure 1 shows the volume fraction of the fluids at different times when CellsPerDirection=128.

Figure 1: Volume fraction of the fluids at different times with CellsPerDirection=128

Spatial convergence

In order to study the convergence of the mesh, five spatial discretizations were used.