## 1 Introduction  
Flows around circular cylinders and spheres have been attracting the interest of researchers for more than a century owing to their relatively easy and controllable conditions, as well as the interesting flow physics [1; 2]. These problems have often been considered as prototypes of flows around bluff bodies which are relevant to engineering and practical applications. In these cases, the body geometry is described by a single parameter (the diameter \(d\) of the cylinder or of the sphere) and the flow is characterized only by the Reynolds number \(Re_{d}=Ud/\
u\), with \(U\) the free stream velocity and \(\
u\) the kinematic viscosity of the fluid.  
A possible extension of the above flow paradigms consists of considering a geometry that is described by two curvature parameters instead of one. We believe that in this way it is possible to establish the standard for a new benchmark that, even if yet to be well defined and sufficiently simple, shows a richer and more complex flow physics with respect to circular cylinders and spheres.  
In the present paper we have considered the flow around a toroidal ring with radius \(R\) and core diameter \(2a\) in an attempt to increase the complexity of the problem. In fact, it is possible to change, in addition to the Reynolds number \(Re=RU/\
u\), also the slenderness parameter \(\Gamma=2a/R\). Apart from the paper by [3] that used a rarefied hypersonic flow, this problem has been considered for an incompressible flow by several studies in the past. Reference [4] considered \"fat\" tori (\(\Gamma=2\)) in the Stokes regime, while [5] investigated only slender tori (low \(\Gamma\)) in the very low Reynolds number range. Reference [6] conducted experiments to measure the drag in axisymmetric conditions, and [7; 8; 9] performed accurate and extensive numerical simulations and stability analyses showing that all the dynamics from the flow around a sphere (\(\Gamma\to\infty\)) to that around a circular cylinder (\(\Gamma=0\)) can be obtained by varying the cylinder slenderness.  
In this study we have analyzed the low Reynolds number range \([10^{-2};3\times 10^{3}]\), and the aspect ratio has been spanned from \(\Gamma=0.1\) up to the limit \(\Gamma=2\); the results are summarized in the phase diagram of Figure 1, confirming that indeed the aspect ratio \(\Gamma\) plays a fundamental role in the determination of the flow features. The novel finding of this study is the existence of a new regime in the \(Re\)-\(\Gamma\) phase space, indicated by \(\mathbf{H}\) in Figure 1. In this region the flow develops a spherical vortex in the wake detached from the surface of the torus, whose vorticity is distributed as in the Hill spherical vortex [10].  
It is worth mentioning that the phase diagram of Figure 1 has been obtained from the results of about 100 numerical simulations, more than 20 of which are fully three-dimensional. The boundaries between the regions have been somehow arbitrarily set by observing the flow behaviour and gathering the simulations into homogeneous categories. Therefore, the lines dividing the different regions should not be interpreted as sharp fronts, but rather as regions where the transition occurs.  
The paper is organized as follows: in Section 2, the numerical setup is briefly described, the problem is introduced, and the results are validated. Section 3 contains the results with an analysis of the various flow regimes. The paper is closed by Section 4 with the conclusions.