<p dir="ltr">The increasing availability of spatio-temporal information in the form of detailed time-resolved images sampled at very high framing rates has resulted in a search for mathematical techniques capable of extracting and relaying the pertinent underlying physics governing complex flows. Analysis relying on the usage of a solitary spectral, correlation, or modal decomposition techniques to identify dynamically significant information from large datasets may give an incomplete description of these phenomena. Moreover, fully resolved spatio-temporal measurements of these complex flow fields are needed for a complete and accurate description across a wide spectrum of length and time scales. The primary goals of this dissertation are address these challenges in two key aspects: (1) to improve and demonstrate the novel application of complementary data analysis and modal decomposition techniques to quantify the dynamics of flow systems that exhibit intricate patterns and behaviors in both space and time, and (2) to make advancements in achieving and characterizing high-resolution and high-speed quantitative measurements of turbulent mixing fields.</p><p dir="ltr">In the first goal, two canonical flow fields are considered, including an acoustically excited co-axial jet and a bluff-body stabilized flame. The local susceptibility of a nonreacting, cryogenic, coaxial-jet, rocket injector to transverse acoustics is characterized by applying dynamical systems theory in conjunction with complementary wavelet-based spectral decomposition to high-speed backlit images of flow field. The local coupling of the jet with external acoustics is studied as a function of the relative momentum flux ratio between the outer and inner jets, giving a quantitative description of the dynamical response of each jet to external acoustics as a function of the downstream distance from the nozzle.</p><p dir="ltr">Bluff bodies are a common feature in the design of propulsion systems owing to their ability to act as flame holders. The reacting wake behind the bluff body consists of a recirculation bubble laden with hot-products and wrapped between separated shear layers. The wake region of a bluff body is systematically investigated utilizing a technique known as robust dynamic mode decomposition (DMD) to discern the onset of the thermoacoustic instability mode, which is highly detrimental to aerospace propulsion systems. The approach enables quantification of the spatial distribution and behavior of coherent structures observed from different flows as a function of the equivalence ratio.</p><p dir="ltr">As modal decomposition techniques employ a certain degree of averaging in time, a novel space-and-time local filtering technique utilizing the well-defined characteristics of wavelets is introduced with a goal of temporally resolving the spatial evolution of irregular flow instabilities associated with specific frequencies. This provides insight into the existence of transient sub-modal characteristics representing intermittencies within seemingly stable modes. The flow fields obtained from the same two canonical flows are interrogated to demonstrate the utility of the technique. It has been shown that temporally resolved flow features obtained from wavelet filtering satisfactorily track the same modal featured derived from DMD, but reveal sub-modal spatial distortions or local non-stationarity of specific modal frequencies on a frame-by-frame basis.</p><p dir="ltr">Finally, to improve the ability to study the dynamical behavior of complex flows across the full range of spatio-temporal scales present, advancements are reported in the spatial and temporal quantitative measurement of the scalar quantities in turbulent mixing fields utilizing 100 kHz planar laser-induced fluorescence (PLIF) and Rayleigh scattering imaging of acetone. The imaging system provided a resolution of 55 µm with a field-of-view mapping of 18.5 µm/pixel on the camera sensor, which is three times better spatial resolution than the previous reported work to-date for similar flow fields that were investigated at 1/10^th the current measurement rate. The power spectra of instantaneous mixture fraction fluctuations adhere to Kolmogorov's well-established -5/3 law, showing that the technique captures a significant range of dissipation scales. This observation underscores the ability to study mixing dynamics throughout the turbulent by fully resolving scalar fluctuations up to 30 kHz. This enhanced spatio-temporal resolution allows for a more detailed investigation of the dynamical behavior of turbulent flows with complex modal interactions down to the smallest diffusion limited mixing scales.</p>