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This test case solves advection of a sphere of fluid in a sheared analytical velocity field. It is interesting because of the large variation of the interface topology. It is particularly adapted to measure mass-conservation. The objective is to compare two VOF-PLIC (Piecewise Linear Interface Construction) advection methods, the conservative method presented by [1] and the non conservative one presented in [2].
The simulation takes place in a 3D box of coordinates \( (0,0,0) \) and \( (1,1,1) \). This domain is discretized uniformly with N cells in each directions. A bubble of fluid A is initialized in a domain full of fluid B. The coordinates of the center of the bubble at initial time is \( (0.35,0.35,0.35) \) with a \(0.15m\) radius. The simulation runs during \( T=1.5s \).
The velocity field is inverted at time \( t=T/2 \) such a way to obtain ideally a final state equal to the initial state. For the first \( 0.75s \), the field is defined as :
\begin{align} u(x,y,z) = 2sin(\pi x)^2sin(2\pi y)sin(2\pi z) \\ v(x,y,z) = -sin(2\pi x)sin(\pi y)^2sin(2\pi z) \\ w(x,y,z) = -sin(2\pi x)sin(2\pi y)sin(\pi z)^2 \end{align}
To ensure that CFL of the method is always respected, we set time step \( \Delta t \) as
\begin{align} \Delta t = \frac{0.2}{N} \end{align}
Below is shown a plot of the simulation with a (256,256,256) grid at time \( t=0.75s \).
![Figure 1 : Isocontour of the volume fraction value 0.5 at time t=0.75s, N=256] (single_vortex_3D.png)
In order to evaluate mass loss \( \varepsilon \) during the simulation, we compute the difference between the volume fraction of fluid at initial time and at time T.
\begin{align} \varepsilon = \sum \limits^{N^3}_{i=1}f_i - \sum \limits^{N^3}_{i=1}f^0_i \end{align}
To evaluate mesh convergence, we define the \( L_1 \) norm of the difference between the initial and the current volume fraction such as :
\begin{align} L_1 = \sum \limits^{N^3}_{i=1} |f_i - f^0_i| \\ \end{align}
We also compute local convergence order \( p \) of the method evaluating the evolution of the error with the mesh refining:
\begin{align} p = \frac{ln\left( \frac{E_2}{E_1} \right)}{ln\left (\frac{N_1}{N_2} \right)} \end{align}
where \( E_1 \) and \(E_2 \) are error for mesh 1 and mesh 2, of size \( N_1\) and \(N_2\) respectively.
The following tables show, for different spatial discretization the material loss for the two different advection methods.
\begin{align} \textbf{Non-conservative method} : \end{align}
| Grid size N | \( \Delta x \) | \( \varepsilon_{air}\) (non conservative method) | \( L_1 \) | p |
|---|---|---|---|---|
| 16 | 1/16 | \( -1.59 \times 10^{-3}\) | \( 1.07 \times 10^{-2} \) | |
| 32 | 1/32 | \( -7.57 \times 10^{-4} \) | \( 4.27 \times 10^{-3} \) | \(1.32 \) |
| 64 | 1/64 | \( -3.73 \times 10^{-4} \) | \( 1.63 \times 10^{-3} \) | \(1.38 \) |
| 128 | 1/128 | \( -1.85 \times 10^{-4} \) | \( 4.96 \times 10^{-4} \) | \(2.71 \) |
| 256 | 1/256 | \( -9.26 \times 10^{-5} \) | \( 2.34 \times 10^{-4} \) | \(1.08 \) |
\begin{align} \textbf{Conservative method :} \end{align}
| Grid size N | \( \Delta x \) | \( \varepsilon_{air}\) (conservative method) | \( L_1 \) | p |
|---|---|---|---|---|
| 16 | 1/16 | \( -5.97 \times 10^{-14} \) | \( 1.10 \times 10^{-2} \) | |
| 32 | 1/32 | \( -5.97 \times 10^{-14} \) | \( 4.41 \times 10^{-3} \) | \(1.31 \) |
| 64 | 1/64 | \( -6.66 \times 10^{-16} \) | \( 1.05 \times 10^{-3} \) | \(2.07 \) |
| 128 | 1/128 | \( 9.99 \times 10^{-16} \) | \( 4.93 \times 10^{-4} \) | \(1.09 \) |
| 256 | 1/256 | \( 1.56 \times 10^{-14} \) | \( 1.89 \times 10^{-4} \) | \(1.38 \) |
Results confirm that even though the method presented in [2] shows pretty good results concerning mass conservation, the one presented in [1] is fully conservative (machine precision). Moreover, the convergence order is near to one for both methods and nealy same error is noticed.
[1] G.D. Weymouth, Dick K.-P. Yue, Conservative Volume-of-Fluid method for free-surface simulations on Cartesian-grids, Journal of Computational Physics 229 (2010) 2853–2865
[2] Jérôme Breil. Modélisation du remplissage en propergol de moteur a propulsion solide. Mécanique des fluides [physics.class-ph]. Université de Bordeaux 1, 2001. Français. tel-0147869. https://tel.archives-ouvertes.fr/tel-01478691
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