5 Analysis of a Production FSI ® Cylinder-Head
5.6 POD of Experimental and Numerical Results
5.6.1 Analysis of the Post-Processing Window Size Influence
Following the POD theory, described in §3.3.2.4, the kinetic energy obtained with the different modes has been computed and plotted on Figure 5.23. As commonly employed in POD applications, a “mean-shift” has been realized here, in other words, the averaged mean flow was subtracted from the original data set before performing the POD analysis. This subtraction does not normally alter the form of the POD modes due to the large energy associated with the mean flow. Indeed, a preliminary test based on the present data is coinciding with other observations for similar studies in the literature, where the mean flow is appearing as the highest energy mode, without changing the structure of the following modes. Finally the shift operated ensures that the data are centred on the origin (as advised in [26] to improve the interpretation of the POD results).
0 2 20 18 16 14 12 10 8 6 4
0 2 14
12
10
8
6
4 0
2 20 18 16 14 12 10 8 6 4
0 2 14
12
10
8
6
4
0 2 20 18 16 14 12 10 8 6 4
0 2 14
12
10
8
6
4 0
2 20 18 16 14 12 10 8 6 4
0 2 14
12
10
8
6
4
Figure 5.23: Cumulative energy distribution of the POD modes and relative difference - DES 100C3 (valve plane) results with: a/ the full range of modes and b/ a zoom in the 30 most energetic modes - c/ and d/
corresponding SAS results
The representation in Figure 5.23 offers an overview over the convergence of the modes resulting from the decomposition process. The comparison of the energy contained in the 300 modes chosen for this analysis is showing a different behaviour towards the windowing effect between DES and SAS results. Apparently, DES results are more sensitive to the window size than the SAS ones. Indeed, in graphs a/ and b/, one can observe that the difference between the full valve-plane and the restricted window is reaching its maximum around the 20th mode at about 2% with the DES model and then decreasing slowly with the amount of modes taken into account, down to a value of almost 1% after 100 modes. For the SAS model results, graphs c/
and d/, this difference stays below 1% at its maximum, which is around the 50th mode and is already below 0.5% after 100 modes. Another diverging point between these two models is the amount of energy contained in the first mode. In the DES decomposition, the first mode contains approximately 7.5% of the total fluctuant kinetic energy, whereas in the SAS case the same mode is representing more than 10% of the total energy. However, on a more global perspective, beside the light discrepancies mentioned before, the two different post-processing windows are providing very similar results for each model.
Energy [%]
Mode
Energy [%]
Mode
a/ b/
Energy [%]
Mode
Energy [%]
Mode
c/ d/
Full plane
+ Experiments window Difference
Full plane
+ Experiments window Difference
DES
SAS
Figure 5.24: Cumulative sum of the kinetic energy up to the truncation mode indicated in each graph normalized with the total kinetic energy resulting from the 300 modes
Figure 5.24 above shows the sum of the kinetic energy obtained thanks to the truncation mode indicated in the upper left hand side corner, normalized with the total energy resulting while considering the maximum number of modes (in the present case, 300 modes). The ratio defined
as
∑ ∑
=
= N
j j Nt
i
i k
k
1 1
/ , where Nt is the truncation mode and N =300, offers another representation of the convergence of the different modes. It is then possible to associate the different flow regions with a faster or slower energy convergence. For example, from the plot previously mentioned, one can observe a relatively fast convergence in the core of the cylinder or in the flow region located under the flow splitter in the intake port. These regions are also the most quiescent regions, where the flow is most stable. On the opposite, positions with high shear stress or undergoing large fluctuations, such as around the valve-shaft or starting from the valves tip and along the wake of the plate, are the last regions to converge. This aspect is representative of the high complexity of the flow developing in the cylinder head, where the large number of degrees of freedom, or in other words the dimension of the problem, requires taking into account a large number of modes to reproduce properly the phenomenon.
Another interesting aspect is observed between modes 40 and 50, in the right hand side of the valve-shaft where the energy is suddenly converging within less than ten modes. A more detailed
0 1 0.75 0.5 0.25 0 1 0.75 0.5 0.25
m1 m10
m40 m50
m100 m200
analysis will shed light on the precise modes and frequencies at which this rapid change is occurring.
The following figures (Figure 5.25 to 5.28) are showing the topology of the spatial modes (shaded with the kinetic energy) obtained for the DES results, classified by POD level from the most energetic, large scale structures, mode 1 in Figure 5.25, to the smaller ones (here until the fourth mode, Figure 5.28). While comparing the two different post-processing fields, apart from the first mode showing high similarities in both planes, some clear discrepancies are appearing for the next modes. Although the energy distribution plotted formerly in Figure 5.23 is really comparable in both cases, the spatial modes are showing some differences. On a global aspect, some similarities exist with regards to the resolved structures at each mode, but their location does not always match with each other. A more quantitative analysis of the spatial structures corresponding to the different modes will be presented later in this chapter.
Figure 5.25: Visualization of the 1st POD mode of the flow (DES100C3) – Velocity vectors and kinetic energy: a/ full valve cross-section plane b/ PIV window
a/ b/
75 100
50 25 0
[m²/s²]
Figure 5.26: Visualization of the 2nd POD mode of the flow (DES100C3) – Velocity vectors and kinetic energy: a/ full valve cross-section plane b/ PIV window
Figure 5.27: Visualization of the 3rd POD mode of the flow (DES100C3) – Velocity vectors and kinetic energy: a/ full valve cross-section plane b/ PIV window
a/ b/
a/ b/
75 100
50 25 0
[m²/s²]
75 100
50 25 0
[m²/s²]
Figure 5.28: Visualization of the 4th POD mode of the flow (DES100C3) – Velocity vectors and kinetic energy: a/ full valve cross-section plane b/ PIV window
Even though the window dimension is influencing, in a certain extent, the shape of the individual modes, the capacity to reconstruct the original velocity field is lying on the combination between spatial modes and their associated time coefficients. Indeed, the structures resulting from the decomposition cannot be considered individually as real structures existing in the flow, but just constitute an optimal N-dimensional basis to reconstruct the original data set.
However, as it will be seen from the following paragraph, by reconstructing the instantaneous velocity fields according to the same truncated modes, both windows are delivering identical results. Although the individual modes are not perfectly fitting together, the reconstruction based on the combination of spatial modes fluctuating in accordance with their time coefficients is giving back the original data set. Thus, the size of the window used for POD should not affect the quality of the decomposition and the reliability of the reconstructed information.