## 4 Results and Analysis
### Spectral Characteristics of Jet Space and Near Wall  
The spectrum analysis of the jet space and near wall is expected to obtain an intrinsic evolution mechanism of the coherent structure. The dimensionless parameter of the frequency (\(f\)) is the Strouhal number (\(St\)), which can be represented as  
\[St=f\times D/U_{b} \tag{17}\]  
Figure 11 shows the power spectral density of the axial fluctuating velocity in jet space at \(x\)/\(D\) = 0, \(y\)/\(D\) = 1.7 and \(x\)/\(D\) = 0.5, \(y\)/\(D\) = 1.7. Two positions are located in the potential flow core region and axial shear layer, respectively. It can be seen that frequency attenuation in these regions contains a \(-\)5/3 energy spectrum region for the jets issuing from the 0\({}^{\circ}\) straight hole and 45\({}^{\circ}\) spiral hole. Due to the action of the vortices in the shear layer of the jet, the magnitude of the power spectral density in the shear layer is higher than that in the potential flow core region. It is indicated that there is an inertia sub-region in the simulation of turbulent flow field, but the interval length of the region is relatively short. Following the inertia sub-region, the power spectral density decays rapidly, which may be related to the truncation effect of the filter in the turbulent model. A common characteristic for non-swirling jets is the distinct formation of periodic structures in the axial shear layer. The natural frequency corresponds to the axi-symmetric vortex rolling up in the vicinity  
Figure 10: The mean second invariant in jet space at \(y\)/\(D\) = 0.05 along the \(x\) direction.  
of the nozzle outlet, and primary vortices are alternatingly shed with the sub-harmonic frequency in the vortex pairing process. The swirling flow issuing from the 45\({}^{\circ}\) spiral hole is weak with a swirling number of about 0.18, and the power spectra for the u-velocity component for the non-swirling and swirling issue from the differences of two holes exit in the sub-harmonic frequency. For the non-swirling and swirling impinging jets, both of the natural frequencies are about 1, and the sub-harmonic frequencies are 2.18 and 1.15, respectively. It can be noted that as swirl is applied, the sub-harmonic and fundamental frequencies become less distinct, which may be due to the interaction of the azimuthal mode instability with the base flow instabilities at the common actions of azimuthal and axial shears.  
Figure 12 shows the power spectral density of radial fluctuating velocity at different locations of \(y\)/\(D\) = 0.05 near the target surface. Overall, both of the change trends are consistent for the non-swirling and swirling jets. As swirl is applied, power spectral densities become slightly higher at the downstream region. In Figure 12a, the radial velocity fluctuating at the stagnation point is quite weak, the power spectrum is lower in two orders of magnitude compared to the downstream region of other positions, and the sub-harmonic frequency is not distinct. In Figure 12b, for the jet of the 45\({}^{\circ}\) spiral hole, the wall jet occurs earlier along the wall and the radial fluctuating velocity is larger at \(x\)/\(D\) = 0.5; that is, the power spectrum peak of the radial fluctuating velocity has a smaller frequency. For the jet of the 0 \({}^{\circ}\) straight hole, airflow near the wall at \(x\)/\(D\) = 0.5 begins to deflect gradually and enters the wall-free region. The scale of the airflow vortex is smaller and unstable, which results in two distinct frequencies of the power spectrum. In Figure 12c, airflow near the wall enters the region of the wall free jet for the jets of the 0\({}^{\circ}\) straight hole and the 45\({}^{\circ}\) spiral hole; these are two distinct frequencies of the power spectrum. In Figure 12d, only the natural frequency of the power spectrum exists. After the development of the wall free jet, the radial velocity near the wall dominates the wall boundary layer and gradually stablises.  
Figure 11: Power spectral density of the axial fluctuating velocity in jet space.