# 2D Airfoil

## Problem definition

This test case consists in the simulation of the flow around a stalled ellipsoid at a Reynolds number $latex Re=5000$. With this test case we aimed to assess the hybrid method’s capability to simulate flows with higher Reynolds number and to evaluate its performance in a situation where vortices exit and re-enter the Eulerian sub-domain.

The configuration of the hybrid domain is presented below:

The Lagrangian sub-domain, $latex \Omega_{L}$, covers the whole fluid domain and the Eulerian domain, $latex \Omega_{E}$, is restricted to the vicinity of the solid boundary. The parameters used in this simulation are presented in the next table: ## Results

The contour plots of vorticity, comparing the hybrid results to the FE ones, are presented below: Plot of the vorticity field for $latex t \in \{1.0, 2.0, 3.0, 3.5, 4.0\}$, comparing the hybrid simulation (left) with the FE simulation (right).

We can see that the two solvers give very similar results up to $latex t=3s$ and after that they start to diverge. This behaviour is expected due to the non-linear nature of the problem. Nevertheless we can see a comparable vortical structure. Below we plot the contour plots of vorticity for the hybrid solver only for an extended period: Plot of the vorticity field for $latex t \in \{1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0m 9.0, 10.0\}$ computed by the hybrid solver.

Regarding the drag and lift, we can observe that the hybrid solver is capable of reproducing very well the FE results up to $latex t=2s$, see below: Evolution of the lift coefficient from $latex t=0$ up to $latex t=10$. The figure compares the hybrid results with FE simulation. Evolution of the drag coefficient from $latex t=0$ up to $latex t=10$. The figure compares the hybrid results with FE simulation.

After that instant, which corresponds to the start of the vortex shedding, the results stop having a good match, but remain within the same bounds and show similar behaviour.