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Simulations with isolated bodies
The objective of this part is to reproduce the simulations presented in the thesis [Thesis_Berti]. We consider two-dimensional rigid bodies placed in a box filled with incompressible Newtonian fluid. These rigid bodies are subjected to gravity and therefore fall towards the bottom of the box. When one rigid body reaches the bottom, the simulation is stopped.
1. Falling disk
For this first test, the rigid body is a disk. The parameters defining this disk, as well as the box and the fluid are given by :
Name |
Description |
Values |
Unit |
\(\Omega\) |
Computational domain (box) |
[\(0,2\)] \(\times\) [\(0,6\)] |
\(cm\) |
\(R\) |
Disk radius |
\(0.125\) |
\(cm\) |
\(D\) |
Disk diameter |
\(0.25\) |
\(cm\) |
\(O\) |
Disk center |
(\(1,4\)) |
\(cm\) |
\(\rho_s\) |
Disk density |
\(1.5\) |
\(\frac{g}{cm^3}\) |
\(\rho\) |
Fluid density |
\(1\) |
\(\frac{g}{cm^3}\) |
\(\mu\) |
Fluid viscosity |
\(0.01\) |
\(\frac{cm^2}{s}\) |
\(g\) |
Gravity acceleration |
\(980\) |
\(\frac{cm}{s^2}\) |
\(\Delta t\) |
Simulation time step |
\(10^{-3}\) |
\(s\) |
We visualize the time evolution of the vertical position of the disk’s center of mass \(y^{CM}\) and compare these positions with the obtained results from the thesis.
One can observe that we obtain the same results.
We redo the same simulation by changing the disk density and the fluid viscosity. The new values of these parameters are given by :
Name |
Description |
Values |
Unit |
\(\rho_s\) |
Disk density |
\(1.25\) |
\(\frac{g}{cm^3}\) |
\(\mu\) |
Fluid viscosity |
\(0.1\) |
\(\frac{cm^2}{s}\) |
For this simulation we represent, besides the evolution of the vertical position of the center of mass of the disk, the time evolution of the vertical velocity, of the Reynolds number and of the kinetic energy.
The vertical velocity \(U_y(t)\) is defined by :
\[U_y(t) = \frac{y^{CM}(t) - y^{CM}(t-\Delta t)}{\Delta t}.\]
The expression of the Reynolds number \(Re(t)\) and the kinetic energy \(E_k(t)\) are respectively given by :
\[Re(t) = \rho_{s} D \frac{ \sqrt{U_{x}^{2} (t) +U_{y}^{2} (t) } }{\mu} \quad \mbox{ and } \quad E_k(t) = 0.5 \rho_{s} \pi R^{2} (U_{x}^{2} (t) +U_{y}^{2} (t) ).\]
The results are shown by the following four figures:
We find for the four quantities the same values as those presented in the thesis. The higher viscosity of the fluid implies that the disk needs more time to reach the bottom of the box.
2. Tilted falling ellipse
The second test consists in an ellipse whose long axis is initially rotated by an angle \(r\) relative to the horizontal. We will visualize the trajectory of the center of mass of this ellipse as a function of time. The used parameters are defined by :
Name |
Description |
Values |
Unit |
\(\Omega\) |
Computational domain (box) |
[\(0,0.4\)] \(\times\) [\(0,2.8\)] |
\(cm\) |
\(L\) |
Characteristic size of the domain |
\(0.4\) |
\(cm\) |
\(a\) and \(b\) |
Ellipse long and short axis |
\(0.1\) and \(0.05\) |
\(cm\) |
\(O\) |
Ellipse center |
(\(0.2,2.4\)) |
\(cm\) |
\(r\) |
Ellipse rotation |
\(45\) |
˚ |
\(\rho_s\) |
Ellipse density |
\(1.1\) |
\(\frac{g}{cm^3}\) |
\(\rho\) |
Fluid density |
\(1\) |
\(\frac{g}{cm^3}\) |
\(\mu\) |
Fluid viscosity |
\(0.01\) |
\(\frac{cm^2}{s}\) |
\(g\) |
Gravity acceleration |
\(980\) |
\(\frac{cm}{s^2}\) |
\(\Delta t\) |
Simulation time step |
\(10^{-3}\) |
\(s\) |
To visualize this trajectory of the ellipse center of mass, we represent its vertical position \(y^{CM}\) as a function of its horizontal position \(x^{CM}\), both normalized by the characteristic size of the domain \(L\). The figure is given by :
We also obtain the same results for this benchmark.
References on fluid - body interaction
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[Thesis_Berti] Luca Berti (2021). Numerical methods and optimisation for micro-swimming.
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[Maury] B. Maury (1999). Direct Simulations of 2D Fluid–Particle Flows in Biperiodic Domains. Download PDF