## 3 Results
### Physics of the Side-Jets Formation Phenomenon  
#### 3.3.1 Generation of Longitudinal Streamwise Vortices  
The mechanism of side-jet formation was studied experimentally by [PERSON] and [PERSON] [6] and numerically by [PERSON] et al. [9]. In both cases, the number and orientation of the side-jets were fixed by an external jet perturbation. In the experimental analysis [6], stabilisation of the side-jets at fixed azimuthal angles was achieved by a corrugated nozzle, which caused deformation of the vortex structures at specific locations. In the DNS [9], the deformation of the vortex rings was introduced by application of an azimuthal perturbation. In both studies, the side-jets originated from a velocity induction by pairs of counter-rotating streamwise vortex structures, which connect the two consecutive distorted vortex rings. The same mechanism of the side-jet formation is revealed in the present simulations in both the Cartesian and cylindrical coordinate system. Figure 8 presents the iso-surfaces of the Q-parameter, contours of the streamwise vorticity and the radial velocity with the velocity vectors in the cross-section perpendicular to the jet axis obtained for the test cases with the lowest turbulence intensity (\(E-Car-Ti_{0.01}\) and \(E-Cyl-Ti_{0.01}\)). The \(Q\)-parameter exhibits the formation of pairs of streamwise vortical structures. Sometimes, they are called rib or braid vortices, and their occurrence is frequently reported in studies of non-circular jets, e.g., rectangular, triangular or hexagonal jets [32]. There, the rib vortices originate from local disturbances introduced by the corners of the nozzles. In the present case of circular jets, the rib vortices are generated by a slightly different mechanism. A careful analysis of the results presented in Figure 8a shows that there are four pairs of ribs. If one plots them with the computational mesh in the background, one could readily see that the ribs are generated along the diagonal directions of the mesh, and the azimuthal distance between neighbouring pairs is 90 deg. Hence, it is inferred that the disturbance that triggers the formation of rib vortices is introduced by an azimuthal non-uniformity of the computational mesh. Indeed, at angles of 45, 135, 225 and 315 deg, the spacings between the mesh nodes along the radius of the jet are the largest, and thus, the accuracy of approximation in these directions is the worst. In effect, the generated ribs are persistent in the same localisations during the whole simulation. Knowing the origin of perturbation in the Cartesian coordinate system, one may ask \"What is the mechanism of the formation of braids when the computations are performed in the cylindrical coordinate system?\" The iso-surfaces of the \(Q\)-parameter in Figure 8b clearly show the pairs of rib vortices, but their localisation and number have nothing common with the mesh structure and seem accidental. Indeed, as will be shown later when the cylindrical coordinate system is used, the number of ribs and their spatio-temporal occurrence can be regarded as a random process conditioned by the randomness of the inlet velocity perturbation. If at a given instant in time and at a given azimuthal angle in the inlet plane of the cylindrical mesh the generated velocity disturbance has a tendency to grow faster than the disturbances in the neighbouring localisation, it creates rib vortices.