Higher order dynamic mode decomposition of noisy experimental data: The flow structure of a zero-net-mass-flux jet

Abstract

A method is presented to treat complex experimental flow data resulting from PIV. The method is based on an appropriate combination of higher order singular value decomposition (which cleans the data along the temporal dimension and the various space dimensions) and higher order dynamic mode decomposition (HODMD), a recent extension of standard dynamic mode decomposition that treats the data in a sliding window. The performance of the method is tested using experimental data obtained in the near field of a zero-net-mass-flux (ZNMF) jet. The better performance of HODMD is put in evidence making this technique suitable to both, cleaning the experimental noise using a limited number of snapshots and obtaining robust and sufficiently accurate results that elucidate the spatio-temporal structure of the flow. The results show that this ZNMF jet is temporally periodic in the near field, where the flow results from the interaction of a large number harmonics. These harmonics involve large scale spatial flow structures, identified as spatially growing instabilities, which are associated with the flow transition to turbulence in the far field.

A Reduced Order Model to Predict Transient Flows around Straight Bladed Vertical Axis Wind Turbines

“…Nevertheless, the solution obtained would improve when using a larger value for d, if more snapshots are retained. The reason is that the HODMD expansion, characterized by d reduces data uncertainty [24] and helps removing the transient modes, which are treated by the method as noise [13]). Regarding the tolerances, Figure 4 shows that there are small variations of velocity, of order 10 −2 (velocity fluctuations), which represent variations of 10% with respect the mean velocity.…”

“…This tolerance is determined, among other factors, by the level of noise [24,29] or the level of the fluctuations of the signal [30].…”

“…After some calibration, based on the robustness (similar solutions for various type of calculations) of the results [24,29], the parameters used for the HODMD analysis are 5 ≤ d ≤ 10, ε 1 = ε 2 = 10 −3 . It is remarkable that, although the flow is highly complex and the analysis is carried out in an unsteady regime of a numerical simulation (only ∼1 blade revolution is analysed), which limits the values of d used for the analysis [29], the results are robust.…”

“…Higher Order Dynamic Mode Decomposition (HODMD) [11] is an extension of the Dynamic Mode Decomposition (DMD) approach [23], which has been recently introduced as a tool to analyze complex and noisy flows [24] or data whose spatial dimension is restricted to external conditions (i.e., hot-wire measurements or flight flutter test experiments [25]). The DMD approach extracts modes (or flow structures) from complex flows, which are ranked by frequency.…”

“…The parameter d is set by the user, through a calibration phase based on the robustness of the results (for various ε 1 , ε 2 , ∆t and various d). The final results should become independent of these parameters or, in other words, need to be robust to small variations of these parameters [24].…”