## 2 Problem and Numerical Setup
The computational domain is discretized by a mesh which is non-uniform in the radial and axial directions to cluster the gridpoints in the torus region and in the wake (see the example in the sketch of Figure 2); for the three-dimensional cases, the mesh is uniform in the azimuthal direction. The axisymmetric cases have been mainly run on two different grids \(120\times 201\) and \(258\times 401\) in the radial and axial directions, respectively, obtaining a drag coefficient \(C_{D}\) (computed by integrating viscous stresses and pressure over the torus surface and projecting the normalized force in the streamwise direction) that typically showed a difference below \(1\%\). In many cases--especially those with low \(\Gamma\) and high \(Re\)--a third grid of \(515\times 801\) nodes has been used to further assess the grid independence of the results again with differences below \(1\%\), which is within the statistical uncertainty of the data. The _standard_ domain dimensions are \(L_{r}=16R\) and \(L_{x}=50R\), with the inflow distance \(L_{i}=10R\); however, since the ideal flow consists of an _isolated_ torus, we have verified that the domain finiteness did not significantly affect the results through the blockage effect. In more detail, the cases at \(\Gamma=1\) for \(Re=100\) and \(Re=200\) have also been run using a second domain of dimensions \(L_{x}^{\prime}=90\) and \(L_{r}^{\prime}=30\), and the number of nodes augmented accordingly: the differences in the drag coefficient now below \(0.15\%\). In a second test it has been verified that the torus was placed far enough from the inflow to have results independent of \(L_{i}\): this was checked by repeating the runs at \(\Gamma=1\) and \(Re=100\) and \(Re=200\) with \(L_{i}^{\prime}=20R\) and on a grid extended ahead of the torus; in this case, the maximum difference of the drag coefficient was below \(2\%\), which is acceptable for our purposes. Owing to the large CPU requirements, only a few three-dimensional cases were run, and all of them on the _standard_ domain with a grid \(258\times 401\times 65\) in the radial, axial, and azimuthal directions. These last runs were mainly aimed at showing the onset of three-dimensional instabilities rather than analyzing the flow details. A first validation of the results is given by the analytical study of [4], who computed the Stokes flow around a \"closed\" torus (\(\Gamma=2\)) for which they found a drag coefficient \(C_{D}\simeq 5.611/Re\); the present numerical results yield \(C_{D}=5.63\) at \(Re=1\) and \(C_{D}=56.04\) at \(Re=0.1\), which are in very good agreement with the prediction. A further validation is provided in Figure 3 showing the drag coefficient (i.e., the force in the opposite direction with respect to the mean flow normalized by the factor \(\rho U^{2}S/2\), with \(S\) the blockage area) as function of the Reynolds number. The results are compared with the classical paper by [13], and the deviations range from a minimum of \(0.53\%\) at \(Re=100\) up to a maximum of \(1.50\%\) at \(Re=200\). It is worth mentioning that the validation in the range \(100\leq Re\leq 1000\) is much more challenging than that at low Reynolds, owing to the steeper velocity gradients in the flow and the thinner boundary layers at the sphere surface. Further validation is shown in the next section for different aspect-ratios \(\Gamma\) and Reynolds number ranges. Here it suffices to mention that the results of Figure 3 confirm that the numerical tool is reliable and the criteria for the selection of the run parameters (grid spacing and time step) are adequate. Figure 2: Sketch of the problem.