## 3 Results
### Non-Excited Jets
#### 3.1.2 Instantaneous vs. Time-Averaged Results  
Figure 5 shows the contours of instantaneous axial velocity distributions in the central cross-section in the Cartesian and cylindrical coordinate systems obtained for \(Ti=0.01\%,0.1\%\) and \(1.0\%\). These data are presented on the left sides of the subfigures. The presented values are normalised by the inlet velocity in the axis, i.e., \(\langle U_{x}\rangle/U_{0}\). It can be seen that the instantaneous values locally exceed the \(U_{0}\), which is the result of the vortices' rollup process in the shear layer of the jet, as one can see in Figure 3. The counterclockwise rotation of the toroidal vortices causes the acceleration of the fluid in the central part of the jet. A close analysis of the presented results shows the regions where \(\langle U_{x}\rangle>U_{0}\) are located in between these vortices. Note that these regions occur further downstream when the level of turbulence intensity is low. In effect, larger values of the time-averaged axial velocity extend further downstream. Its contours are shown on the right-hand sides of the subfigures presented in Figure 5. Evidently, with decreasing \(Ti\), the potential core length (\(L_{PC}\)), i.e., the region in the axis where \(\langle U\rangle\approx U_{0}\), becomes longer. Taking into account the solutions obtained in the Cartesian and cylindrical coordinate systems, it can be seen that a substantial difference in \(L_{PC}\) is only seen for the cases with \(Ti=0.01\%\). Figure 4: Evolution of the spectra of the axial velocity fluctuations registered at the jet axis. The results obtained for \(NE-Car-Ti_{0.01}\) and \(NE-Cyl-Ti_{0.01}\). Figure 5: Contours of instantaneous (**left sides** of the subfigures) and time-averaged (**right sides** of the subfigures) values of the axial velocity \(U_{x}/U_{0}\) in the central cross-section in the Cartesian and cylindrical coordinate systems obtained for \(Ti=0.01\%,0.1\%\) and \(1.0\%\). Profiles of the time-averaged axial velocity and its fluctuation along the axis are shown in Figure 6. Additionally, to verify the correctness of the present solutions, the experimental and numerical data from [PERSON] et al. (2016), [PERSON] and [PERSON] (2017), and [PERSON] and [PERSON] (2018) are presented in Figure 6. These data were obtained for \(Ti=1.0\%\) at the Reynolds numbers \(\mathrm{Re}=4050\)([PERSON], 2016) and \(\mathrm{Re}=10,000\)([PERSON], 2016; [PERSON], 2018). It can be seen that both the rate of velocity decay in the region \(x/D>5\) and the potential core length are predicted accurately. The profiles of the fluctuations also agree relatively well with the literature data, especially for the case with the same turbulence intensity (\(Ti=1.0\%\)). The occurrence of the maximum of the fluctuations is correctly captured, yet its level is slightly over-predicted. This, however, can result from differences in spatio-temporal turbulent length scales in the present simulations and the ones prescribed in the cited references. Taking into account the impact of \(Ti\), it is clear that its alteration changes the \(L_{PC}\). It equals \(L_{PC}/D\approx 3.5\) for \(Ti=1.0\%\) and \(L_{PC}/D\approx 5.5\) for \(Ti=0.01\%\). Note that the \(Ti\) level also affects the growth of velocity fluctuations along the axis. In all cases, they start increasing before the potential core regions ends, but this happens further downstream when \(Ti\) is low. As a consequence, the maxima of the fluctuations shift downstream as well. Regarding the impact of the coordinate system on the obtained results, it can be seen that for the highest and moderate turbulence intensities, i.e., \(Ti=1.0\%\) and \(Ti=0.1\%\), the solutions are very weakly dependent on the mesh topology, and the profiles nearly overlap each other. Similar to the instantaneous solutions, the time-averaged values differ significantly when \(Ti=0.01\%\). On the Cartesian mesh, the profile of the axial velocity obtained for \(Ti=0.01\%\) is similar to the one computed for \(Ti=0.1\%\).