MOF test case for two fluids with sheared velocity field in the axisymmetric coordinate system

This test case simulates two fluids in a disc box in the axisymmetric coordinate system.

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 centred at the position (0.5,0.75).

Fluids' properties

Both of the fluids considered have the same properties :

\(\rho\)	\(\mu\)	\(\sigma\)	C	\(\alpha\)	\(T_{0}\)
1	1	1	1	1	1

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: No matter the fluid properties we choose, they don't affect the results because we solve only the Phase advection equation.

Velocity

For the velocity field, the sheared case is considered. The velocity is set as: \(u(x,y)=(x \cos(2 \pi y) \sin(2 \pi x) \cos( \dfrac{\pi t}{T} ),- \dfrac{1}{\pi} \sin(2 \pi y) \sin(2 \pi x)+ \pi x \cos(2 \pi x) \cos( \dfrac{\pi t}{T} ) )\) where \(T \) is the final time.

Phase advection

The numerical method considered for Phase advection is MOF.

Runtime parameters

Spatial discretization

We consider the variable CellsPerDirection in order to change the regular mesh. \(\dfrac{L}{\Delta x}=CellsPerDirection\) for each of the grid directions. The default value is 32.

Time discretization

The time step is fixed and set as : \( \dfrac{Final \ time}{Time \ iterations}\).

UseSymmetric choice

The UseSymmetric variable is used to toggle the symmetric reconstruction. The tolerance angle and derivative are respectively set to \(10^{-5}\) and 0.

UseFilaments choice

The UseFilaments variable is included to toggle Filaments at MOF method.

Comments

Inconvenients results are found when filaments are disabled;
Symmetric reconstruction has no big affection on results quality but gives better results in terms of convergence.

Results

Numerical results

Figure 1 shows the volume fractions at different times when filaments are enable, symmetric reconstruction is not used and \(CellsPerDirection=128\). Figure 2 presents the volume fractions when filamnts are disable.

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

Figure 2: Volume fraction of the fluids at different times with CellsPerDirection=128 and without using filaments

Spatial convergence

Four spatial discretizations were used to study the spatial convergence.

Four cases are considered to observe the impact of filaments and symmetric reconstruction:

Filaments and symmetric reconstruction are not used:

Mesh	Area of symmetric difference	order
16×16	0.009893327962988079	n/a
32×32	0.003918579700884963	1.336
64×64	0.001342459515070065	1.545
128×128	0.0003227490898693112	2.056

Filaments are disable and symmetric reconstruction is enable:

Mesh	Area of symmetric difference	order
16×16	0.00956574588046014	n/a
32×32	0.003996532273886688	1.259
64×64	0.001443059198513254	1.470
128×128	0.0004293919100915999	1.749

Filaments are enable and symmetric reconstruction is disable:

Mesh	Area of symmetric difference	order
16×16	0.0036020431989792576	n/a
32×32	0.0006859171862338258	2.393
64×64	0.00015402378264769235	2.155
128×128	4.333096244707189e-05	1.830

Filaments and symmetric reconstruction are enable:

Mesh	Area of symmetric difference	order
16×16	0.0036034247723255074	n/a
32×32	0.0006825542186226577	2.400
64×64	0.00014798179844643082	2.206
128×128	4.3109131213725456e-05	1.779