In this paper, we combine the spectral proper orthogonal decomposition (SPOD) and bispectral analysis to identify the dominant quadratic interaction triads in nonlinear flows, and also extract the corresponding bispectral modes involved in the quadratic interactions. The main idea of this approach is using n ranks of SPOD modes to reconstruct the Fourier realizations of flow data, calculating the bispectrum with the definition of integral measure bispectrum, and then taking the SPOD modes as the flow structures involved in quadratic interactions. This research is an alternative to the bispectral mode decomposition method (BMD). This SPOD-based bispectral analysis method is more time-saving compared with BMD method. Since the SPOD-based bispectral analysis method combines the SPOD and bispectral analysis, it can retain the properties of both SPOD and bispectral analysis. The SPOD inherits the optimality of POD, it can provide a hierarchy of modes in terms of energy, and it is distinct from the spatial POD in the sense of giving spatial-temporal-coherent modes. The bispectral analysis has been widely used for identifying quadratic interactions in the area of statistical analysis. These properties of the SPOD-based bispectral analysis approach can give us advantages to construct the reduced-order model of nonlinear flows which can reproduce the nonlinear dynamics in a more computationally efficient way. Also, the results given by this approach can help us interpreting the energy transfer mechanisms in nonlinear flow. To show the application of this approach, we first apply the approach on several artificial test datasets to check the feasibility and denoising ability of this approach. We find that the SPOD-based method has a good capability of resisting the effect of white noise. Then, we apply it to the data of two real fluid mechanics datasets, the direct numerical simulation (DNS) data of a cylinder flow at Re=500 and the particle image velocity (PIV) data of a Mach 0.6 open cavity flow. We compare the results of the SPOD-based approach with the results of bispectral mode decomposition (BMD). The outcomes given by these two approaches are very similar, which means that the SPOD method can be an alternative to the BMD method for important triads identification and flow structures extraction. We also show that in some cases, more than 1 ranks of SPOD modes are needed for more accurate representation in terms of both the flow energy and bispectrum value.

Particle image velocimetry (PIV) is a non-intrusive optical technique to measure thevelocity in fluid flows by relying on a tracer particles seeded inside of the flow, however this method has various issues that could compromise data acquisition and cause regions of missing data, or gaps: irregular seeding, light reflecting off surfaces, light path obstruction, poor particle choice, and imaging system restrictions. To reconstruct these missing regions, Nekkanti and Schmidt [8] developed the Gappy Spectral Proper Orthogonal Decomposition (Gappy SPOD), as a spectral counterpart to the commonly used Gappy Proper Orthogonal Decomposition (GPOD). This thesis presents improvements made to the Gappy SPOD that allow for application to raw PIV data, including a detailed gap finding method and processing of black zones (regions missing in all snapshots). We then compare this method with MF GPOD, and Nearest Neighbor (NN) interpolation in terms of their turbulent kinetic energy (TKE) error in both a global and local region capacity. We find that Nearest Neighbor often outperforms Gappy SPOD or GPOD in regions containing few missing elements. This performance suffers in regions where gap size exceeds 10^3 elements and Gappy SPOD and MF GPOD provide more accurate results. The raw data sets examined here contain greater amounts of small gaps, thus the proposed hybridization of these methods in this thesis will lead to greater accuracy.

Turbulent jets are canonical flows that occur when fluid emerges from an orifice into the surrounding environment, such as the jet from aircraft engines. As the fluid emerges from the nozzle, it forms an unstable shear layer that grows very rapidly, forming large-scale coherent structures, which are the main sources of aft-angle jet noise. The mechanism behind the generation of jet noise is still not fully understood. Further insights into characteristics of coherent structures can aid our understanding of turbulence, and in modeling and controlling various mechanisms. %The development of techniques for the education of coherent structures is another objective of this work.The main foci of this work are: (i) performing high-fidelity numerical simulations of turbulent jets and extracting physical insights from coherent flow structures, and (ii) developing techniques that extract these flow structures from the large dataset generated by these simulations.

In recent years, spectral proper orthogonal decomposition (SPOD) has emerged as a major tool for extracting coherent structures. In the first part, we extend SPOD for low-rank reconstruction, denoising, prewhitenening, frequency–time analysis, and gappy-data reconstruction. Two approaches for flow-field reconstruction are proposed, a frequency-domain approach, and a time-domain approach. A SPOD-based denoising strategy is also presented, which achieves significant noise reduction while facilitating drastic data compression. A convolution-based strategy is proposed for frequency-time analysis that characterizes the intermittency of spatially coherent flow structures. When applied to the turbulent jet data, SPOD-based frequency–time analysis reveals that the intermittent occurrence of large-scale coherent structures is directly associated with high-energy events. Lastly, a new algorithm, gappy-SPOD, is developed that leverages the space-time correlation of SPOD modes to estimate missing data. Even for highly chaotic flows with up to 20\% missing data, our method facilitates that structures associated with different time scales are well-estimated in the missing regions. For the cases considered here, it outperforms established techniques such as gappy-POD and Kriging.

In the second part, we investigate the nonlinear dynamics and controllability of coherent structures by actuating them. Large-eddy simulations (LES) of two unforced and four forced jets at $Re = 50,000$ and $M_j = 0.4$ were performed. The two unforced jets include an initially laminar and a turbulent jet. All four forced jets are turbulent and are forced at the azimuthal wavenumbers $m=0$, $m=\pm1$, $m=\pm2$, and $m=\pm6$. The unforced and forced jets were validated with companion experiments. Compared to the turbulent jet, the initially laminar jet develops later but at a faster rate, which is a result of the vortex pairing in the shear layer. The emphasis of the analysis is on characterizing the vortex pairing and the associated nonlinear energy transfer. Here, for the first time, we evaluate the spectral energy budget based on the leading modes of the SPOD. Our analysis reveals that energy flows from the fundamental to its subharmonic, resulting in the growth of the subharmonic. These results provide evidence for a previously suggested parametric resonance mechanism. In the forced jets, we examine the effect of forcing using a recently proposed method, bispectral mode decomposition (BMD), which extracts flow structures associated with nonlinear triadic interactions. We use BMD to construct a cascade of triads and find that the most dominant triads arise due to fundamental self-interaction and second-harmonic-fundamental difference interaction. Furthermore, our analysis of the far-field in the unforced and $m=0$-forced jets sheds light on the crucial role of difference-interactions in the generation of jet noise.

Triadic interactions are the fundamental mechanism of energy transfer in fluid flows and arise due to the quadratic non-linearity of the Navier-Stokes equation. The bispectral mode decomposition (BMD) is a direct means of revealing flow structures that are associated with these interactions and accomplishes this by computing modes generated by the interaction of two separate modes. The mode bispectrum, summed mode spectrum, and bispectral modes are the primary results produced through the use of the BMD, with local maxima in the mode bispectrum and summed mode spectrum representing the most prominent triadic interactions of the flow.

However, these results are often difficult to analyze due to uncertainty in determining what constitutes a local maxima. In this work, a radial search algorithm that has been developed is applied to both the mode bispectrum and summed mode spectrum of open cavity flow data to find their respective maxima. Once found, these maxima are reconstructed into two new visual aids, the sparse mode bispectrum and an arrangement of the bispectral modes. The sparse mode bispectrum condenses the results of radially searching the mode bispectrum and summed mode spectrum, while local maxima found through the radial search of the mode bispectrum have their corresponding bispectral modes arranged in the relative shape of the mode bispectrum. These visualizations aim to present the most important information and results from the BMD according to the radial search algorithm, allowing for the identification of the most pertinent flow physics while also easing the use of the BMD.

This work delves into three fundamental aspects of fluid dynamics: (\romannum{1}) theoretical investigation, (\romannum{2}) computational methodologies, and (\romannum{3}) reduced-order modeling.The starting point is the near-perfect model for general fluid flows: the Navier-Stokes equations. For the first part, we investigate the motion of point vortices under the assumption of small Mach number $(M\ll 1)$. We use a Rayleigh--Jansen expansion and the method of Matched Asymptotic Expansions to analyze the motion of the vortices at different time scales. Our study shows the motion undergoes modifications over long time scales $O(M^2 \log{M})$ and $O(M^2)$.

Generally, the Navier-Stokes equations require the application of accurate, high-fidelity computational methods for their solutions. To this end, we explore the use of radial basis function-based finite difference (RBF-FD) discretizations for both flow simulations and hydrodynamic stability analysis, which comprise the core of the second part. Polyharmonic spline functions with polynomial augmentation (PHS+poly) are used to construct the global differentiation matrices and the discrete linearized Navier-Stokes operators on scattered nodes. A systematic parameter study is carried out to identify a set of parameters that guarantee stability while balancing accuracy and computational efficiency. Based on this, we develop a mesh-free semi-implicit fractional-step Navier-Stokes solver that uses a staggered node layout. We employ classical linear stability (LST) analysis and state-of-the-art resolvent analysis (RA) to identify flow instabilities.

An alternative way to extract large-scale coherent structures from flow-field data is the utilization of modal decomposition techniques. In the third part, we revisit the connection between spectral proper orthogonal decomposition (SPOD) and other techniques while demonstrating its theoretical correspondence to time-delay analysis. Using SPOD modes, we establish two model order-reduction techniques, namely the operator-based Galerkin projection and the data-driven time-delay Koopman approach. Following the core concept of Koopman theory that an infinite-dimensional linear operator can describe the nonlinear dynamics, we inflate the linear state with an exogenous forcing to account for the nonlinear interactions and background turbulence. Closure is achieved by modeling the remaining residue as stochastic noise. The result models accurately predict the initial transient dynamics and reproduce the second-order statistics of broadband turbulent flows.