## 3 Results
### Hill Regime  
For \(\Gamma=\mathcal{O}(1)\), some values of the Reynolds number gave rise to a special steady state which, to the knowledge of the authors, has never been reported before. More in detail, at \(\Gamma=1\) the flow at \(10\leq Re\leq 70\) showed the appearance of a recirculation bubble in the wake of the torus, the bubble consisting of a vortex ring (Figure 6a,b). The shape of this ring is approximately spherical at its onset, but it becomes more elongated in the streamwise direction for increasing \(Re\) owing to its interaction with the strain field produced by the torus. In fact, this recirculation shows strong similarities with the Hill spherical vortex [10] that is characterized by a linear azimuthal vorticity distribution \(\omega_{\theta}=Ar\), with \(r\) the radial coordinate and \(A\) a constant. In view of this similarity, we have verified whether this relation also holds for the recirculation of the torus, and two representative results are shown in Figure 7. It is clear that at \(Re=10\), when the vortex is approximately spherical and not strained by the flow, the linear relationship is attained very closely (note that in Figure 7 the axes do not have the same range in both panels). On the other hand, at \(Re=50\) the bubble is significantly elongated in the streamwise direction (Figure 4c) and partially stripped by the external strain field; accordingly, the \(\omega_{\theta}(r)\) relation shows evident deviations from linearity--especially in the upstream part of the vortex that is more subject to the wake of the torus.  
As a successive step, we analyze how the recirculation is generated: it is expected that the appearance and disappearance of the vortex is somehow related to the relative proportions of fluid passing over and through the torus. As measure of the flux, we have computed the Stokes streamfunction \(\psi(r,x)=\int_{0}^{r}\omega u_{x}(\sigma,x)\mathrm{d}\sigma\) that, computed for \(x=L_{i}\) and \(r=R-a\), yields the flow passing through the torus (inner flux). Some results are reported in Figure 8, showing an increase of the flux with \(Re\) and a tendency to saturation at \"high\" Reynolds number. It is also evident that this flux decreases for increasing \(\Gamma\), since the torus tends to be \"fat\" and the surface of the central hole diminishes. Figure 6d clearly shows that when this flux is vanishing the bubble attaches to the torus as for the separation behind a bluff body. On the other hand, for low values of \(\Gamma\), the inner flux is too  
Figure 6: Instantaneous snapshots of streamwise velocity (gray scales) and Stokes streamfunction (for positive and \(\cdot\cdot\cdot\cdot\cdot\cdot\) for negative values) for the flow around a torus: (**a**) \(\Gamma=1\), \(Re=10\); (**b**) \(\Gamma=1\), \(Re=25\); (**c**) \(\Gamma=0.5\), \(Re=25\); (**d**) \(\Gamma=1.5\), \(Re=25\).  
intense to allow for an axial steady separation. Therefore, it turns out that this regime can only exist for some combinations of \(Re\) and \(\Gamma\) to have an inner flux strong enough to produce a recirculation (through the adverse pressure gradient downstream of the torus) but not too strong to wash it away. The \(Re\)-\(\Gamma\) values for which this special steady state was found are contained within the region **H** of Figure 1.