## Dipole in an unbounded domain

### Problem definition

This test case deals with the evolution of a vortex dipole in an unbounded domain. With this test case we intend to investigate how the flow solution produced by the hybrid solver is perturbed as it traverses the Eulerian sub-domain. In order to do this, we used as initial condition the Clercx-Bruneau dipole, [Clercx et al. 2006], with a positive monopole at $latex (x_{1}, y_{1}) = (-1.0, 0.1)$ and negative monopole at $latex (x_{2}, y_{2}) = (-1.0,-0.1)$, each having a core radius $latex R = 0.1$ and characteristic vorticity magnitude $latex \omega_{e} = 299.528385375226$ as given by [Renac et. al. 2013],

$latex

\omega(x,y,0) = \omega_{e}\left(1-\frac{r_{1}^{2}}{R^{2}}\right) e^{-\frac{r_{1}^{2}}{R^{2}}} – \omega_{e}\left(1-\frac{r_{2}^{2}}{R^{2}}\right) e^{-\frac{r_{2}^{2}}{R^{2}}}\,,

$

where $latex r_{i}^{2}=\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}$. The Eulerian sub-domain is defined as $latex \Omega_{E}=[-0.25,0.25]\times [-0.5,0.5]$, meaning that the dipole is initially placed to its left, as depicted in {fig::cb_convection_schematics}.

### Results

We ran a simulation with the hybrid solver, using the parameters presented in the following table

We then compared it to the results obtained with the FE solver. A contour plot of vorticity, comparing the hybrid results to the FE ones is shown below.

Below we plot the time evolution of the maximum of vorticity, showing the transfer of information between the two sub-domains of the hybrid solver.

In the following figure, we compare the evolution of the vorticity maximum for different particle sizes, $latex h=5.0\times 10^{-3}$ and $latex h=1.0\times10^{-2}$.

As can be seen, for larger particle sizes it is possible to notice a strong decrease in the vorticity maximum as the dipole enters the Eulerian sub-domain. This produces a small reduction in the mean propagation speed of the dipole, as can also be seen in the contour plots above, where the hybrid solution is slightly lagging behind the FE one.

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## Dipole collision with wall

### Problem definition

To further investigate the properties of the hybrid flow solver, namely its ability to capture the generation of vorticity at a solid wall, we applied it to the collision of the Clercx-Bruneau dipole with a solid wall, [Clercx et al. 2006]. A FE solution was validated against the results of Clercx and Bruneau, [Clercx et al. 2006], and used as reference.

The setup of the hybrid domain is as shown below

The Eulerian sub-domain, $latex \Omega_{E}$, covers the near-wall region and the Lagrangian sub-domain domain extends to the complete fluid domain, which is bounded by the no-slip wall $latex \Sigma_{w}$. The parameters used in this simulation are shown in the following table:

### Results

In the next figure we show contour plots of vorticity, comparing the hybrid results to the FE ones.

We can see a good agreement between the two results. In the following figures we compare the evolution of vorticity maximum:

energy, $latex E$:

enstrophy, $latex \Omega$:

and palinstrophy, $latex P$:

As can be seen, the hybrid solver is capable of reproducing the results of the FE solver, with only small variations on the energy and palinstrophy.

## [fruitful_sep]

## References

Clercx, H. J. H., & Bruneau, C.-H. (2006). The normal and oblique collision of a dipole with a no-slip boundary. *Computers & Fluids*, *35*(3), 245–279. doi:10.1016/j.compfluid.2004.11.009

Renac, F., Gérald, S., Marmignon, C., & Coquel, F. (2013). Fast time implicit–explicit discontinuous Galerkin method for the compressible Navier–Stokes equations. *Journal of Computational Physics*, *251*, 272–291. doi:10.1016/j.jcp.2013.05.043