Numerical methods and computational approaches for studying fluid flow have become increasingly popular and more mature in their capabilities since the 1960s, when the prohibitive experimental costs associated with flight vehicle development programs necessitated an alternative approach. Computational Fluid Dynamics (CFD) approaches for several practical applications were pioneered in the early 1980s, but despite the algorithmic advances the computing power necessary to compute full three-dimensional flow-fields remained a bottleneck. With the advent of powerful microprocessors and high performance supercomputers, high fidelity three-dimensional CFD for cases of practical interest have become feasible over the last couple of decades. Although these high fidelity simulations contain the desired physics, extracting that information is proving to be a challenge due to their extremely large size. For instance, a 3D Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) of an airfoil can produce terabytes of data. For such large datasets, it becomes difficult to use conventional analysis and visualization techniques without resorting to statistical methods more suited to such ``big data" problems. There are several such methods, each of which usually places emphasis on a certain aspect of the dataset. However, these can be used in conjunction with each other to complement each other to generate a comprehensive understanding of the fundamental physical mechanisms prevalent in a flow-field.
The central goal of this dissertation is to develop strategies based on novel statistical, model reduction and signal processing techniques to derive such physical insights into large CFD datasets of practical interest. To demonstrate this, three high fidelity LES datasets of various unsteady flow-fields arising in Micro Air Vehicle (MAV) flight have been analyzed in depth. They are a) Static stall of a NACA 0015 airfoil with plasma control, b) Dynamic stall in a plunging SD 7003 airfoil, and c) Interaction of a stream-wise oriented vortex impinging on a rectangular wing, which is a canonical problem in the aerodynamics of formation flight.
First, model reduction as an effective means to study complex, realistic flows is investigated. The recently popularized model reduction technique, Dynamic Mode Decomposition (DMD) has been given special emphasis in addition to the well known Proper Orthogonal Decomposition (POD), to study the three cases mentioned above. In the NACA 0015 static stall case, DMD analysis reveals a strong instability in the flow-field at a Strouhal number St = 2. Subsequent actuation with a Nano-Second Pulse Dielectric Barrier Discharge (NS-DBD) plasma actuator at this frequency is found to mitigate stall, indicating the effectiveness of DMD analysis in estimating actuation parameters for flow control a priori. The potential of DMD in understanding the physics of a turbulent flow-field is then further explored using the SD 7003 airfoil dynamic stall case. DMD shows that the large scale energetic structures in the leading edge vortex oscillate with a frequency equal to that of the airfoil plunging frequency, with several harmonics.
Additionally, the relationship between DMD and POD modes are also studied, showing how key similarities among these different techniques can be leveraged to form a more perceptive understanding of the physics.
Apart from model reduction for physics extraction, a major emphasis of this work has been on analysis of non-stationary signals arising from turbulent flows. Despite Fast Fourier Transform (FFT) and DMD being effective tools to study local and global periodic features in the flow, the inherently non-stationary nature of turbulence in several applications pose significant limitations to their reliable usage. Thus, such datasets require special treatment to account for their non-stationary behavior, with reduced approximation errors. One such technique that has been explored in this work is the Empirical Mode Decomposition (EMD). EMD was initially popularized in the earth sciences community to study intermittent features and trends in highly non-stationary signals. The effectiveness of EMD in fluid mechanics is demonstrated by analyzing the case of a stream-wise oriented vortex impinging on a rectangular wing, where EMD extracts the prominent surface pressure signatures from temporal signals of the impinging vortex. Furthermore, it is shown that these results can be used in conjunction with DMD, such that key local frequencies identified by EMD guide DMD to extract their global surface pressure footprint on the airfoil surface. Some more recent works in literature have also pointed to the utility of EMD in analyzing non-stationary data in turbulence, and its adoption is likely to increase in the research community.
Although EMD proves to be a useful solution for feature extraction from non-stationary signals, it has some significant practical limitations. A crucial impediment is that EMD in its basic form can be applied only for univariate/single-channel data. For multivariate signals, the algorithm cannot easily extract common patterns and features in the different channels, due to theoretical constraints in its mathematical formulation. This is a significant deficiency in fluid mechanics since signals from several sources, in both experiment and CFD, are often used to study the flow, intrinsically making it a multivariate dataset. Case-specific bivariate and trivariate analysis for turbulence have not shown much promise.
To address this need, a generalized n-variate treatment, Multivariate EMD (MEMD) is adopted as a potential solution in this work. MEMD is a very recent development which has been primarily used in the neuroscience community to analyze multichannel data. Its application in turbulence has been demonstrated in this work by using it to study the inception (or onset) of dynamic stall in the SD 7003 case mentioned above. The stall inception data exhibits a very high degree of non-stationarity across time and space. Since MEMD, unlike EMD, successfully extracts intermittent features/oscillations among the multivariate signals, it is able to uncover several new dynamics of the flow which were previously elusive.
Finally, this work also extends its focus to the spectral analysis of non-stationary signals. Although the Fast Fourier Transform (FFT) has been widely used to study spectra in turbulent datasets, they tend to be deficient in generating the spectra for non-stationary signals where the frequency/wavenumber often varies. This is because these methods can reliably extract only periodic features in the signal, while the other features tend to be missed. Wavelet analysis is a very popular and time-tested alternative mathematically well suited to compute transient spectra for non-stationary signals, and is widely used in different domains of science and engineering. However, they require a user defined parameter known as the mother wavelet, the choice of which significantly impacts their effectiveness.
Often, careful selection of the mother wavelet requires some a priori insight into the signal. This can be a problematic, especially during exploratory data analysis of turbulent flows.
As a potential solution, this work proposes that the algorithm of Matching Pursuits (MP), which has been typically used in biomedical sciences for spectral analysis and sensor fusion, to compute spectra. MP needs minimal user guidance for this purpose, and is therefore very robust and repeatable. The application of MP to analyze spectra during the inception of stall demonstrates excellent performance. Furthermore, the stall inception case has been used to show that MEMD and MP can be used in tandem to track and analyze disturbances in the flow. The results indicate that inception of dynamic stall may be a two-stage process, with the spatio-temporal movement of various flow structures dictating the dynamics of each stage. Therefore, this approach provides a general framework to study other problems in fluid mechanics where rapidly evolving flow features may need to be tracked.