## 3 Results
### Unsteady Regime  
A further increase of the Reynolds number enhances the inertial effects with respect to the viscous ones, and this induces flow transition from steady to unsteady. The recirculations that in the steady regime were attached to the torus are now alternatively shed in the wake and advected by the main flow. A typical case is shown in Figure 10 for the aspect ratio \(\Gamma=0.1\) and \(Re=1000\), and it is similar to the flow around a circular cylinder.  
A characteristic quantity of the unsteady vortex street is the Strouhal number \(St=Rf/U\), where \(f\) is the main frequency of the flow. Here \(f\) has been measured from the first peak of the fast Fourier transform of the drag coefficient (Figure 11a), even if any other quantity (for example, velocity or pressure sampled in a point) yielded the same result. Similarly to the Reynolds number, the relation \(St_{d}=St\Gamma\) has been used to compare the present results with those of a circular cylinder, since the Strouhal number for a cylinder \(St_{d}\) is computed using the cylinder diameter. Figure 11b shows the normalized oscillation frequencies of the flow as a function of the Reynolds number, and in order to make the comparison easier, all the data are normalized in the same way as for the circular cylinder. For \(\Gamma=0.1\), the Strouhal number for the torus appears to be close to the corresponding value for the cylinder, yet systematically below it. Similarly to the separation length of Figure 9a, also in this case the difference is due to the finiteness of the aspect ratio and this is confirmed by the analogous quantities for \(\Gamma=0.25\) and \(\Gamma=0.5\) showing larger discrepancies and in the same direction as the case at \(\Gamma=0.1\)  
Figure 9: (**a**) Length of the separation bubble \(l_{S}\) normalized by the core diameter versus Reynolds number: \(\circ\) torus at \(\Gamma=0.1\), filled \(\triangle\) circular cylinder (from [10]). Instantaneous snapshots of streamwise velocity (gray scales) and Stokes streamfunction (\(\underline{\phantom{\circ}}\) for positive \(\dot{\  
Another fundamental difference between the wake of a cylinder and that of a torus is that the vortices of the latter are in fact vortex rings, and they have a self-induced translation velocity that adds (algebraically) to the convection of the wake. For a vortex ring with thin core and uniform vorticity distribution within the core, the self-induced translation velocity is \(V=\gamma/(4\pi b)[\ln(8b/\epsilon)-1/4]\), where \(\gamma\) is the core circulation, \(b\) the ring radius, and \(\epsilon\) the core radius. Looking at Figure 10, we can see that the vortices shed from the external side rotate clockwise while the others rotate counterclockwise; this implies that the internal vortices must travel downstream faster than the external ones, and accordingly the wake tends to diverge radially as shown by the straight lines connecting the vortex centres. This behaviour is different from that of the cylinder, whose wake spreads symmetrically with respect to the streamwise direction.  
In the case at \(\Gamma=0.1\), the internal and external vortices have very similar dimensions, and the measured differences of \(\gamma\) were below 2%. However, the situation becomes different as \(\Gamma\) increases because the external vortices increase in size at the expense of the inner vortices that eventually disappear. This completely changes the instability mechanisms, and tends to inhibit the flow unsteadiness. Figure 11b shows that indeed for a given \(Re\) the oscillation is slower for increasing \(\Gamma\), and we have found that for \(\Gamma>1\) the axisymmetric unsteadiness is suppressed with a direct transition from steady axisymmetric to three-dimensional flows (Figure 1).