Proyecto 5 (Complejo de apartamentos)

The snapshots method provides, in principle, a number of modes equal to the number of snapshots (in this test case, 500 realizations). For clarity, in Figs. 8.16- 8.18-8.20 the plot is limited only to the first 100 modes, containing the bulk of the energy. The eigenvalues are normalized with respect to their sum, representing the total turbulent energy of the fluctuations. Furthermore, the cumulative sum of the energy is reported, in order to identify the number of modes that significantly contribute to build up the decomposition of the velocity field.

In Fig. 8.16 the case of no upstream swirl is illustrated. Evidently, the first three modes constitute the most relevant contribution, as they contain respectively 20.8%, 10.9% and 7.5% of the energy, respectively (building in total 39.2% of the overall energy). The modes 4-9, contain in total about 10% of the energy; the modes 10-500 are associate each one with less than 1% of the total energy, thus they are most certainly associated with random fluctuations due to small scale turbulence and measurement noise.

The first three modes for the case of are reported in Fig. 8.17, in which the contour representation of the component, normalized with the bulk velocity, and the iso-surfaces of and are reported. The first and second modes are associated with the large scale precession, as they present an asymmetric outflow associated with an inflow on the opposite side of the chamber. This interpretation is confirmed by observing the instantaneous velocity fields reported in Fig. 8.12. The strong periodic component of the flow field enables the possibility to reconstruct the main topology of the precessing motion using only the first two modes (see Sec. 8.4.3). The difference in the energy pertaining to the first and the second mode is probably related to the geometry of the measurement volume, since it contains the entire inflow and outflow regions for the case of the first mode, while in the second mode the same two highly energetic regions are partly located outside of the observed volume in the far field. The third mode is quasi-axisymmetric, as it presents a strong on-axis outflow, associated to the axial mode. As a matter of fact, the ratio of the energy associated to the first two modes and the total energy of the first three modes can be interpreted as an “effective energy – probability” of precessing motion (about 81% for the case of ).

Chapter 8 – Jet flows past a sudden expansion

Fig. 8.16 Energy distribution for the case of : a) normalized eigenmodes; b) cumulative energy.

Fig. 8.18 Energy distribution for the case of : a) normalized eigenmodes; b) cumulative energy.

Fig. 8.19 Contour representation on slices of the , and iso-surfaces of and for the first (a), second (b) and third (c) POD mode for the case of .

Chapter 8 – Jet flows past a sudden expansion

Fig. 8.20 Energy distribution for the case of : a) normalized eigenmodes; b) cumulative energy.

Also in the case of the first three modes contain a significant part of the energy, i.e. 11.7%, 6.6% and 3.5% of the energy, respectively, as outlined in Fig. 8.18. In this case the energy of the first three modes constitutes only 21.8% of the total energy, mainly due to the effect of swirl, which determines a faster and stronger mixing, thus spreading part of the energy over a wide spectrum of modes. The first three modes are illustrated in Fig. 8.19 with the same layout of Fig. 8.17. The results confirm the presence of two dominant modes induced by the precession, and a third mode associated with the axial outflow. The analysis of the eigenvalues determines a precession probability of about 84%, which is only slightly larger than the case of the circular jet. This result indicates that the upstream swirl does not bias significantly the flow field towards a stable precessing motion with respect to a non swirling inlet.

As expected, the POD for the case of presents significantly different features with respect to the two previously presented cases. A quite limited part of the energy is contained in the first four modes (3.2%, 2.9%, 1.9%, 1.5%, constituting about 9.5% of the overall energy, see Fig. 8.20). The spreading of the energy over the set of modes is due to the strong mixing effects of the vortex breakdown, determining an extremely fast decay of the turbulent energy along the spectral pipeline. Nevertheless the physical content of the first modes is still significant. Differently from the cases of and , the first two modes (Fig. 8.21) present both an inflow and an outflow separated along the direction, the main difference being the width of this separation (that is much smaller for the second mode). The second mode is related to a far-field precession effect induced by the precessing vortex core, while the first mode is similar to some extent to those of and , but with inflow and outflow confined in the exit region. This suggests a more significant interaction with the external field, which goes beyond the topic of this investigation. The third and the fourth mode are relative to the recirculation region dominating the near field of the sudden expansion.

Fig. 8.21 Contour representation on slices of the , and iso-surfaces of and for the first (a), second (b), third (c) and fourth (d) POD mode for the case of .

Chapter 8 – Jet flows past a sudden expansion

Fig. 8.22 Scatter plot of the time coefficients for the first two modes in the normalized plane. The

circumference with radius 1 is plotted for reference.