In this section, two modal decomposition methods are compared to analyze the numerical database recorded at the chamber/turbine interface of the two-component LES. The objective is to better understand the global dynamics of the flow and, in particular, to study the effect of the PVC on the properties of the flow at the interface. Additional information of the flow topology for this configuration are widely available in the literature (see, for example, [
22,
34]), but it is well known that this configuration features a PVC frequency around 500 Hz. It is expected that the modal decompositions will be able to exhibit such dynamics. Two modal decompositions are compared in the following. spectral proper orthogonal decomposition (SPOD) is first considered as it is able to provide orthogonal modes in a spatiotemporal domain, which is particularly suited for this configuration, showing a strong time dependency linked to the PVC. The second method is the standard proper orthogonal decomposition (POD), which produces modes orthogonal in a space-only domain.
2.2.1. Modal Decomposition 1: Spectral Proper Orthogonal Decomposition (SPOD)
To begin with, the use of SPOD is considered. Unlike POD, SPOD provides modes that are both spatially and temporally coherent (and not just spatially for POD). A more detailed analysis of the characteristics of the SPOD is given in [
8]. Thanks to this property of spatiotemporal coherence, the SPOD modes are able to decouple the structures of the flow with different time scales. This aspect is particularly interesting from an analysis point of view for a configuration such as the one studied here. Indeed, the PVC frequency that dominates the dynamics of the flow can be easily isolated in the spectral domain, given its important contribution to the total energy of the flow. The spatial structure of this dynamic is thus accessible without effort. This observation is also valid for other frequencies, in particular, for the first harmonics of the PVC.
To perform a modal decomposition of the database using SPOD, the fluctuations of each snapshot of the flow for the density, the three velocity components, and the static temperature are stored in a vector
. The scalar product between two snapshots is chosen to be equivalent to the total energy norm [
35,
36]. Following the explanations given in [
35], the eigenmode decomposition of the cross-spectral density matrix is estimated using Welch’s method from an ensemble of snapshots. To do so, the initial database containing a total of 20,000 snapshots is divided into blocks and each block is composed of 3200 snapshots, which gives a set of seven blocks to use in the Welch method. As explained in [
35], the choice of the SPOD parameters (block size and number of blocks) corresponds to a compromise between the spectral resolution and the number of modes.
The SPOD spectrum obtained with these parameters contains a total of seven modes and is presented in
Figure 5. The eigenvalues are normalized by the sum of the total energy on all the frequencies and modes. The blue line corresponds to the power spectral density which is reconstructed by summing the energies of each mode for all the frequencies of the spectrum. Mode 1 clearly evidences the frequency of the PVC at 500 Hz, as well as its first harmonic at 1000 Hz.
At a given frequency, the presence of a significant shift in terms of energy between the first mode and the following modes shows that the modal decomposition is low order, that is to say that the physical phenomenon of this frequency is very well described by the first mode, and the latter dominates the dynamics of the flow. This is mainly the case for the frequency at 500 Hz and for its first harmonic at 1000 Hz. To demonstrate this property of the flow,
Figure 6 plots the eigenvalues at 500 Hz and 1000 Hz as a function of the seven modes, normalized by the total energy at each frequency. It corresponds to the evolution of
along the two dashed red lines of
Figure 5 at frequencies 500 Hz and 1000 Hz. It is clear that at 500 Hz, mode 1 is significantly more energetic than the others since it captures about 80% of the total energy at this frequency. Likewise, this effect is also present at 1000 Hz, where the dominant mode 1 captures more than 30% of the energy.
Moreover, in the SPOD spectrum, the presence of a peak at 500 Hz for the first mode dominating the overall energy level shows that the flow at the chamber/turbine interface is mainly driven by the PVC, which generates high activity at this frequency. On the contrary, when the difference between the eigenvalues of the modes is not significant (here for most frequencies, with the exception of the frequency of the PVC and its harmonics), it means that a single mode is not capable of describing the flow physics. This is, for example, the case for high frequencies, associated with flow turbulence. To describe the physics of the flow at these frequencies, we must instead consider the SPOD modes as a basis of eigenvectors of comparable importance. This behavior can be easily understood, turbulence being a rather random phenomenon, it cannot be described by a unique mode for the corresponding frequencies.
To learn more about the flow topology using the SPOD decomposition, the spatial representation of the first mode for different frequencies is analyzed. The spatial modes of temperature are shown in
Figure 7. At 500 Hz, the first mode clearly shows one lobe at the inner wall and two lobes at the outer wall, in phase opposition. These modes result from the excitation of the flow injected through the multiperforations by the PVC frequency. At 1000 Hz, the same phenomenon is observed, with a doubling of the number of lobes (two external and four internal lobes). For these two frequencies, as explained previously, the first mode is largely dominant in terms of energy compared to the following modes. The dynamics of the flow at these two frequencies is therefore mainly driven by the spatial shape of mode 1. For frequencies between 1500 Hz and 8000 Hz, a progressive decrease in the structure size is observed. This can be explained simply by the fact that smaller structures in the flow are governed by smaller time scales (and thus higher frequencies). To continue the analysis, the spatial representation of the same frequencies for the axial velocity is given in
Figure 8. This time, the flow activity at 500 Hz and 1000 Hz is not restricted to the near-wall region; it extends over the entire height of the channel. This behavior is again related to the PVC. Unlike the temperature, which is rather homogeneous in the channel center, as the injection temperature of the hot flow is constant, the velocity varies at the outlet of the injector due to the swirling motion introduced by the PVC. Therefore, the spatial structures of the modes at these frequencies also extend to the center of the channel. For frequencies between 1500 Hz and 8000 Hz, the observations are similar to those made for temperature. The structures are very homogeneous and the size of the structures decreases as the mode frequency increases.
The use of SPOD to analyze the database of snapshots at the chamber/turbine interface clearly demonstrated its capability to identify the main flow dynamics in terms of frequencies. What follows is the same analysis using proper orthogonal decomposition in order to underline the benefits of SPOD for the considered configuration.
2.2.2. Modal Decomposition 2: Proper Orthogonal Decomposition (POD)
POD was introduced by Lumley [
6,
37] for the analysis of turbulent flows. In practice, POD enables the reduction of an initial database of high dimension to another of lower dimension. The purpose of the method is to determine a base of orthogonal spatial modes, which facilitates the identification of coherent structures in the flow with the objective of producing a reduced model of the initial database. A detailed comparison between the different methods of modal decompositions is made in [
8]. In short, POD provides a base of coherent eigenmodes in space only. The method is therefore not capable of highlighting the temporal correlations which may exist within the database. However, temporal coherence is a very present phenomenon in most flows containing large coherent structures.
In practice, the eigenmodes of POD are composed of coherent structures coming from different frequencies. This results in a wide-frequency spectrum for the temporal coefficients, as demonstrated later. This combination of frequencies for the eigenmodes means that each eigenmode mixes flow physical phenomena of different time scales. As a consequence, it is more difficult to extract and analyze the contribution of different coherent flow structures.
To illustrate the previous remarks, a POD decomposition is performed on the initial numerical database with
(for reasons of memory capacity, only 10,000 snapshots are used instead of the 20,000 available in the full numerical database). The analysis of the 10,000 eigenvalues of the modal decomposition shows that the decrease in terms of energy is very slow (
Figure 9). About 2200 modes are needed to capture 90% of the total energy and over 4600 modes to recover 99% of the total energy. This highlights the fact that POD has difficulty producing a reduced model of the original database.
Another interesting illustration is given in
Figure 10. It represents the power spectral density of each POD temporal coefficient with contour normalized by the maximum value of its power spectral density (PSD). For a given mode
j, this means that for a given frequency along the x-axis, the darker the color, the closer its level is to the PSD maximum of the POD temporal coefficient of mode
j. For example, the red dashed line associated with mode 1 shows that its temporal coefficient mainly contained the frequency associated with the PVC at 500 Hz. This is also the case for mode 2. On the contrary, for the rest of the modes, the large number of frequencies are close to the maximum of the power spectral density. The second plot shows the PSD of mode 100 located along the green dashed line. It is clear that no frequency dominates over the others. This representation highlights the preceding remark: the temporal coefficients of the eigenmodes obtained by POD contain a wide range of frequencies and the associated modes thus represent a behavior of the flow which combine several time scales, which was not the case for the SPOD.