3D Drop benchmark
The previous section described the strategy we used to track the interface. We couple it now to the Navier Stokes equation solver described in [Chabannes]. In the current section, we present a 3D extension of the 2D benchmark introduced in [Hysing] and realised using Feel++ in [Doyeux].
1. Benchmark problem
The benchmark objective is to simulate the rise of a 3D bubble in a Newtonian fluid. The equations solved are the incompressible Navier Stokes equations for the fluid and the advection for the level set:
where \(\rho\) is the density of the fluid, \(\nu\) its viscosity, and \(\mathbf{g} \approx (0, 0.98)^T\) is the gravity acceleration.
The computational domain is \(\Omega \times \rbrack0, T\rbrack \) where \(\Omega\) is a cylinder which has a radius \(R\) and a heigth \(H\) so that \(R=0.5\) and \(H=2\) and \(T=3\). We denote \(\Omega_1\) the domain outside the bubble \( \Omega_1= \{\mathbf{x}  \phi(\mathbf{x})>0 \} \), \(\Omega_2\) the domain inside the bubble \( \Omega_2 = \{\mathbf{x}  \phi(\mathbf{x})<0 \} and stem:[\Gamma\) the interface \( \Gamma = \{\mathbf{x}  \phi(\mathbf{x})=0 \} \). On the lateral walls and on the bottom walls, noslip boundary conditions are imposed, i.e. \(\mathbf{u} = 0\) and \(\mathbf{t} \cdot (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) \cdot \mathbf{n}=0\) where \(\mathbf{n}\) is the unit normal to the interface and \(\mathbf{t}\) the unit tangent. Neumann condition is imposed on the top wall i.e. \(\dfrac{\partial \mathbf{u}}{\partial \mathbf{n}}=\mathbf{0}\). The initial bubble is circular with a radius \(r_0 = 0.25\) and centered on the point \((0.5, 0.5, 0.)\). A surface tension force \(\mathbf{f}_{st}\) is applied on \(\Gamma\), it reads : \(\mathbf{f}_{st} = \int_{\Gamma} \sigma \kappa \mathbf{n} \simeq \int_{\Omega} \sigma \kappa \mathbf{n} \delta_{\varepsilon}(\phi)\) where \(\sigma\) stands for the surface tension between the twofluids and \(\kappa = \nabla \cdot (\frac{\nabla \mathbf{\phi}}{\nabla \phi})\) is the curvature of the interface. Note that the normal vector \(\mathbf{n}\) is defined here as \(\mathbf{n}=\frac{\nabla \phi}{\nabla \phi}\).
We denote with indices \(1\) and \(2\) the quantities relative to the fluid in respectively in \(\Omega_1\) and \(\Omega_2\). The parameters of the benchmark are \(\rho_1\), \(\rho_2\), \(\nu_1\), \(\nu_2\) and \(\sigma\) and we define two dimensionless numbers: first, the Reynolds number which is the ratio between inertial and viscous terms and is defined as : \(Re = \dfrac{\rho_1 \sqrt{\mathbf{g} (2r_0)^3}}{\nu_1}\); second, the Eötvös number which represents the ratio between the gravity force and the surface tension \(E_0 = \dfrac{4 \rho_1 \mathbf{g} r_0^2}{\sigma}\). The table below reports the values of the parameters used for two different test cases proposed in [Hysing].
Tests 
\(\rho_1\) 
\(\rho_2\) 
\(\nu_1\) 
\(\nu_2\) 
\(\sigma\) 
Re 
\(E_0\) 
Test 1 (ellipsoidal bubble) 
1000 
100 
10 
1 
24.5 
35 
10 
Test 2 (skirted bubble) 
1000 
1 
10 
0.1 
1.96 
35 
125 
The quantities measured in [Hysing] are \(\mathbf{X_c}\) the center of mass of the bubble, \(\mathbf{U_c}\) its velocity and the circularity. For the 3D case we extend the circularity to the sphericity defined as the ratio between the surface of a sphere which has the same volume and the surface of the bubble which reads \(\Psi(t) = \dfrac{4\pi\left(\dfrac{3}{4\pi} \int_{\Omega_2} 1 \right)^{\frac{2}{3}}}{\int_{\Gamma} 1}\).
2. Simulations parameters
The simulations have been performed on the supercomputer SUPERMUC using 160 or 320 processors. The number of processors was chosen depending on the memory needed for the simulations. The table Numerical parameters used for the test 1 simulations: Mesh size, Number of processors, Time step, Average time per iteration, Total time of the simulation. summarize for the test 1 the different simulation properties and the table Mesh caracteristics: mesh size given, number of Tetrahedra, number of points, number of order 1 degrees of freedom, number of order 2 degrees of freedom give the carachteristics of each mesh.
h 
Number of processors 
\(\Delta t\) 
Time per iteration (s) 
Total Time (h) 
0.025 
360 
0.0125 
18.7 
1.25 
0.02 
360 
0.01 
36.1 
3.0 
0.0175 
180 
0.00875 
93.5 
8.9 
0.015 
180 
0.0075 
163.1 
18.4 
0.0125 
180 
0.00625 
339.7 
45.3 
h 
Tetrahedra 
Points 
Order 1 
Order 2 
0.025 
73010 
14846 
67770 
1522578 
0.02 
121919 
23291 
128969 
2928813 
0.0175 
154646 
30338 
187526 
4468382 
0.015 
217344 
41353 
292548 
6714918 
0.0125 
333527 
59597 
494484 
11416557 
The NavierStokes equations are linearized using the Newton’s method and we used a KSP method to solve the linear system. We use an Additive Schwarz Method for the preconditioning (GASM) and a LU method as a sub preconditionner. We run the simulations looking for solutions in finite element spaces spanned by Lagrange polynomials of order \((2,1,1)\) for respectively the velocity, the pressure and the level set.
3. Results Test 1: Ellipsoidal bubble
Accordind to the 2D results we expect that the drop would became ellipsoid. The figure~\ref{subfig:elli_sh} shows the shape of the drop at the final time step. The contour is quite similar to the one we obtained in the two dimensions simulations. The shapes are similar and seems to converge when the mesh size is decreasing. The drop reaches a stationary circularity and its topology does not change. The velocity increases until it attains a constant value. Figure~\ref{subfig:elli_uc} shows the results we obtained with different mesh sizes.
4. Bibliography
.

[Chabannes] V. Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles. PhD thesis, Université de Grenoble, 2013.

[Doyeux] V. Doyeux, Modélisation et simulation de systèmes multifluides, Application aux écoulements sanguins, PhD thesis, Université de Grenoble, 2014.

[Hysing] S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman, S. Ganesan, and L. Tobiska, Quantitative benchmark computations of twodimensional bubble dynamics, International Journal for Numerical Methods in Fluids, 2009.