3.3 Results
3.3.2 Stenosis severity effect via POD analysis
The swirling-strength maps of five selected POD spatial modes, i( )x , for each of the three models are shown in Figure 3.5. Recall that the first few modes of the POD analysis are associated with the most dominant features of the flow and carry most of the information of the flow. In Figure 3.5, the first modes of all three models are seen to exhibit large vortices, as demonstrated for example by modes 1, 2, and 10 in the three leftmost columns. A large clockwise (i.e. negative) vortex can be observed in both the 50% model (with maximum center value of about 1550 s-1) and 70% model (maximum of 3150 s-1) and is associated with the interaction of the cross-flow and jet flow (see previous section) generated as a result of the stenosis, in contrast to the normal model. As the spatial mode number increases, the strength and size of the vortices (based on the outermost contour line) decreases, which reflects the characteristic property of higher POD modes being associated with smaller flow structures. The effect of the stenosis on the flow structure can be seen in both the 50% and 70% models: for example, in the intermediate modes of 20 and 40, a concentrated alternating vortex pattern (shown as staggered positive and negative swirling-strength values) is seen downstream in the jet- flow region. Additionally, compared to the 50% model, the 70% model exhibits more swirls that are of larger size and strength. In modes 20 and 40 of the normal model, no structures were detectable of comparable strength to those observed in the 50% and 70% models.
Although POD spatial modes are useful in terms of providing information on the dynamics of the flow, further information such as the different stages of the energy transition can be extracted from the energy spectrum of the flow. The eigenvalues calculated from the auto-correlation matrix of velocity represent the energy content of each spatial mode (Equation (A.4)).
Figure 3.5: Color maps of swirling strength generated for the spatial modes 1, 2, 10, 20, and 40 in the normal (top row), 50% (middle row), and 70% models (bottom row). The color bar was customized to emphasize variation between -1500 and +1500 s-1 to enable visualization of the details across all three models. The actual maximum strengths present in the first mode of each of the 50% and 70% models are -1550 and -3150 s-1 respectively, where the negative indicates clockwise swirl. Contour lines represent |200| s-1 increments, starting from -100 and +100 s-1 for negative and positive rotation respectively.
Figure 3.6a shows the double-logarithmic plot of the fractional energy (Equation (A.10)) spectrum as a function of the mode number, with notable differences between the three models. The value for the first-mode energy was 72% for the 70%-stenosed model, 83% for the 50% model, and 86% for the normal model. The energy spectra of the 50% and 70% models exhibit a distinct first zone (consisting of the first three modes) in which the fractional energy content decreases rapidly, as well as a distinct second zone in which energy decays following a power-law slope of -1.34 and -0.91 for the 50% and 70% models respectively. This second zone extends over approximately 35 modes (from around mode 11 to 46) for both models. After this inertial range, a steeper dissipative sub-range is recognizable.
Figure 3.6: POD energy plots using a total mode number (N) of 460 and full FOV for each of the three models: (a) Double logarithmic plot of fractional energy spectra as a function of mode number, (b) cumulative energy distribution showing cumulative eigenvalues as a function of normalized mode number (n/N).
In the energy spectrum of the normal model, a pairwise occurrence of some of the early eigenvalues can be observed; these eigenvalues have almost equal values and therefore produce a stepwise appearance in the energy spectrum. This characteristic pairwise occurrence of eigenvalues has been reported in previous POD studies of periodic flows [34-36] and is associated with the presence of a traveling wave. The spatial modes within each pair are similar but shifted in the stream-wise direction, and likewise the paired temporal modes are phase-shifted in time. The pairwise occurrence vanishes gradually as mode number increases.
The differences observed in the energy spectra and decay rates translate to the cumulative energy distribution presented in Figure 3.6b. The total energy contained within the first 10% (i.e. 46) of the modes is about 100%, 98%, and 95% respectively for the normal, 50% and 70% models, reflecting the slower rate at which the cumulative energy converges toward the total energy content with increasing stenosis severity. The slower convergence rate indicates more spatial modes are needed to reconstruct the velocity fields. To further assess the level of flow instability, the values of global entropy were compared. The entropy value is derived from the mode energy contents as described in Equation (A.11). By increasing flow instabilities, the value of entropy increases reaching a maximum of 1 for the hypothetical flow in which the total energy is fully sustained throughout and is distributed evenly over all the modes. The entropy value further supports the increasing level of flow complexity as it increased from 0.11 in the normal model to 0.17 in the 50% and 0.26 in the 70% model.
POD temporal modes, a tn( ), for 6 selected modes are depicted in Figure 3.7, for each of the three models. For all three models, the first temporal mode essentially reflects the shape of the flow rate waveform input at the CCA inlet (i.e. Figure 3.1c) since this mode reflects the average flow organization. The second mode also reflects the shape of the average cardiac cycle waveform, however, it is slightly time-shifted with a similar but inverted shape. As the mode number increases, the temporal modes start to show more irregular patterns and higher frequencies of fluctuation, as seen from mode 10 and onwards. These disturbances are associated with rapid velocity fluctuations and indicate a transitional turbulent regime. It is interesting to note from mode 10 that the fluctuations start at earlier modes for the 70% model; alternatively the 50% and normal models still show a regular pattern. From mode 40, the cardiac phase during which the velocity fluctuations are exhibited can be determined: in the 50% and 70% models, this duration starts from the mid-systolic acceleration and persists all the way down to the systolic foot. However, for the normal model, the fluctuating region is more confined to mid- acceleration and mid-deceleration. For the 50% and 70% models, small periods of fluctuation are also notable during 0.5 to 0.7 ms, which is the time interval associated with the increased flow right before the diastolic phase in the cardiac cycle waveform
(Figure 3.1c). As the mode number increases, the disturbances become more persistent and more equally spread throughout the cardiac cycle indicating background fluctuations, e.g. due to system noise. The reduction of disturbances, starting in the systolic phase, is first evident in the normal model (as seen by the systolic gap, at 100-250 ms, in mode 70), then in the 50% model (mode 100) and later in the 70% model (approx. mode 200, not shown).
Figure 3.7: POD (N=460) temporal modes 1, 2, 10, 40, 70, and 100 for normal (blue plots in each set), 50% (red), and 70% (black) models, showing the temporal coefficients, -a(t), as a function of the cardiac cycle.