## 4 Results and Analysis
### Flow Field Analysis
The value of \(Q\) is the second invariant and is defined as  
\[Q=\left(\Omega_{ij}\Omega_{ij}-S_{ij}S_{ij}\right)/2 \tag{15}\]  
where \(S_{ij}\) and \(\Omega_{ij}\) are the symmetric and antisymmetric components of velocity gradient, i.e., \(S_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\right)\), \(\Omega_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}-\frac{ \partial u_{j}}{\partial x_{i}}\right)\). Thus, \(Q\) represents the local balance between shear strain rate and vorticity magnitude. Then, \(Q\) is dimensionless as:  
\[Q_{c}=QD^{2}/U_{b}^{2} \tag{16}\]  
The flow in the free shear layer gradually forms lots of small-scale vortices, and some of these vortices develop, grow up, and come together, and at last the vortex rings form. The vortex ring may also continue growing and linking up into a large-scale vortex ring. For the MCIJ, small-scale vortex rings in the green region around the potential flow have a small diffusion area, and four large-scale blue vortex rings near the wall distribute in four directions, corresponding to the corners of four straight grooves. As the target is a semi-cylinder concave surface, the discharge capacity for the MCIJ is different. It can be seen that two of the four large-scale blue vortex rings are big, and the other two are small. Figure 8: Streamlines and velocity vectors in the XOY and YOZ planes. Figure 9: Instantaneous iso-Q-criterion surface for the MCIJ and SIJ. Small-scale vortex rings are relatively sparse among these large vortex rings. For the SIJ, many moderate-scale vortex rings in the green region around the potential flow have a big diffusion area, and large-scale blue vortex rings near the wall evenly distribute and link into a circle. It can be inferred that the airflow near the wall escapes mainly along four corners for the MCJJ, while the airflow escapes evenly all around for the SIJ. Jet flow enters the region of vortex agglomeration around a dimensionless height \(y\)/\(D\) = 0.6-0.8, and then at the downstream area at about \(y\)/\(D\) = 0.25, the escaped flow enters the region of vortex stretching and deformation near the target surface, and the large vortex ring appears. Figure 10 shows the mean second invariant in the jet space at \(y\)/\(D\) = 0.05 along the x direction. For the MCJ, the connected vortices are prevalent in the range of \(x\)/\(D\) = 0.5-1.5; two large vortex rings appear at the locations of \(x\)/\(D\) = 0.7 and \(x\)/\(D\) = 1.3. For the SIJ, only two large vortex rings appear at the locations of \(x\)/\(D\) = 0.5 and \(x\)/\(D\) = 1.3 due to the significant entrainment and aggregation effects for the swirling jet. Compared to the MCJ, the first large vortex ring from the jet free shear layer has a bigger intensity, while the second large vortex covers a much wider area; hence, the unconnected vortex region occurs. There are several recirculation vortices with weak intensity in the range of the flow acceleration zone due to the entrained recirculation flow; the generated locations and number of these vortices are also different for the MCJJ and SIJ due to the swirl effect.