# 2D Dipole

## 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}. The domain decomposition for the Clercx-Bruneau dipole convection problem, with the positive pole located at $latex p_{+}=(x_1,y_1) = (-1,0.1)$ and negative pole located at $latex p_{-}=(x_2,y_2)=(-1,-0.1)$. (Not to scale)

### 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. Plot of the Clercx-Bruneau dipole at $latex t=[0,0.2,0.4,0.7]$. The figure compares the hybrid simulation (top halves) against the FE only simulation (bottom halves).

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. Evolution of the maximum of vorticity, $latex \max\{\omega\}$, from $latex t=0$ to $latex t=0.7$. The solutions are compared to FE [solid black] and VPM [dashed black] benchmark simulations with characteristics identical to the Eulerian and Lagrangian components of the hybrid simulation. The figure corresponds to a blob spacing $latex h=0.005$.

In the following figure, we compare the evolution of the vorticity maximum for different particle sizes, $latex h=5.0\times 10^{-3}$ and $latex h=1.0\times10^{-2}$. Evolution of the maximum of vorticity, $latex \max\{\omega\}$, from $latex t=0$ to $latex t=0.7$. The solutions are compared to FE [solid black] and VPM [dashed black] benchmark simulations with characteristics identical to the Eulerian and Lagrangian components of the hybrid simulation. The figure depicts the maximum vorticity of the hybrid method with nominal blob spacing $latex h=0.01$ and $latex h=0.005$.

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.

## 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 domain decomposition for the Clercx-Bruneau dipole collision problem, with the positive pole at $latex p_{+}=(x_1,y_1) = (0.1,0)$ and negative pole at $latex p_{-}=(x_2,y_2)=(-0.1,0)$. (Not to scale)

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: Summary of the parameters used in the hybrid simulation of the Clercx-Bruneau dipole wall collision problem.

### Results

In the next figure we show contour plots of vorticity, comparing the hybrid results to the FE ones. Plot of the dipole at $latex t = [0, 0.2, 0.4, 0.6, 0.8, 1]$ (from left to right and top to bottom), comparing the hybrid simulation (left half) and FE only simulation (right half).

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, $latex P$: