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Modal decomposition techniques are showing a fast growth in popularity for their wide range of applications and their various properties, especially as data-driven tools. There are many modal decomposition techniques, yet Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are the most widespread methods, especially in the field of fluid dynamics.
Following their highly competent performance on various applications in several fields, numerous extensions of these techniques have been developed. In this work, we present an ambitious review comparing eight different modal decomposition techniques, including most established methods, i.e., POD, DMD, and Fast Fourier Transform; extensions of these classical methods: based either on time embedding systems, Spectral POD and Higher Order DMD, or based on scales separation, multi-scale POD (mPOD) and multi-resolution DMD (mrDMD); and also a method based on the properties of the resolvent operator, the data-driven Resolvent Analysis.
The performance of all these techniques will be evaluated on four different test cases: the laminar wake around cylinder, a turbulent jet flow, the three-dimensional wake around a cylinder in transient regime, and a transient and turbulent wake around a cylinder.
All these mentioned datasets are publicly available. First, we show a comparison between the performance of the eight modal decomposition techniques when the datasets are shortened.
Next, all the results obtained will be explained in detail, showing both the conveniences and inconveniences of all the methods under investigation depending on the type of application and the final goal (reconstruction or identification of the flow physics). In this contribution, we aim at giving a—as fair as possible—comparison of all the techniques investigated. To the authors’ knowledge, this is the first time a review paper gathering all these techniques have been produced, clarifying to the community what is the best technique to use for each application.
Among other possible applications, this allows extending the stabilization methods already developed for Gauss-Lobatto nodes to Gauss nodes, also ensuring properties such as entropy stability while retaining their improved accuracy
We compare the GMM-based sensor with state-of-the-art alternatives. All methods are integrated into a high-order compressible discontinuous Galerkin solver, where two stabilization approaches are coupled to the sensor to provide examples of possible applications.
The Sedov blast and double Mach reflection cases demonstrate that our proposed sensor can enhance hybrid sub-cell flux-differencing formulations by providing accurate information of the nodes that require low-order blending. Besides, supersonic test cases including high Reynolds numbers showcase the sensor performance when used to introduce entropy-stable artificial viscosity t capture shocks, demonstrating
the same effectiveness as fine-tuned state-of-the-art sensors. The adaptive nature and ability to function without extensive training datasets make this GMM-based sensor suitable for complex geometries and varied flow configurations. Our study reveals the potential of unsupervised machine-learning methods, exemplified
by this GMM sensor, to improve the robustness and efficiency of advanced CFD codes.
A canonical wavy surface exposed to a Mach 2 flow is investigated through high-frequency Pressure Sensitive Paint, Kulite measurements, and shadowgraph imaging. The wavy surface features a compression and expansion region, two shock-boundary layer interactions, and two shock-separation regions.
The unsteady characteristics of the wall pressure and shock angles are presented, demonstrating an increase in the amplitude of the instabilities when traveling through the shock systems. The pressure sensitive paint measurements confirm a two-dimensional flow pattern with small transversal unsteadiness. Higher-Order Dynamic Mode Decomposition and Spectral Proper Orthogonal Decomposition are implemented to dissect the different flow features, revealing several dominant low-frequency and medium-frequency phenomena. The separation region appears at frequencies with Strouhal numbers between 0.01 and 0.2, confirmed by the frequency content in the local pressure measurement using Kulites and the pressure sensitive paint.
The bump geometry gives rise to streamwise pressure gradients similar to those encountered on the suction side of low-pressure turbine (LPT) blades. Under steady inflow conditions, the separated-flow laminar-to-turbulent transition is initiated by self-sustained vortex shedding due to Kelvin-Helmholtz (KH) instability. In LPTs the dynamics are further complicated by the passage of the wakes shed by the previous stage of blades.
The wake-passing effect is modeled here by introducing a harmonic variation of the inflow conditions. Three inflow oscillation frequencies and three amplitudes are considered. The frequencies are comparable to the wake-passing frequencies in practical LPTs. The amplitudes range from 1% to 10% of the inflow total pressure. The dynamics of the separated flow are studied by isolating the flow components that are respectively coherent with and uncorrelated to the inflow oscillation. Three scenarios are identified. The first one is analogous to the steady inflow case. In the second one, the KH vortex shedding is replaced during a part of the inflow period by the formation and release of a large vortex cluster. The third scenario consists solely of the periodic formation and release of the vortex cluster; it leads to a consistent reduction of the separated flow length over the entire period compared to the steady inflow case and would be the most desirable flow condition in a practical application.
As high-pressure-turbines operate at extreme temperature conditions, base bleed can be applied at the trailing edge of the airfoils, enhancing the thermal protection along the trailing edge surface but also disrupting the trailing edge flow and altering the overall aerodynamic pressure losses. The current work explores the potential use of base bleed as a flow control tool to modulate the flow between turbine blade rows.
Through the numerical analysis of a symmetric airfoil immersed in a subsonic flow, the effects that trailing edge ejection has on the base region properties and the downstream flow are evaluated. In particular, previous research constrained to steady blowing is now extended to consider an unsteady pulsating base bleed injection. Three injection frequencies are investigated, covering a wide range of base bleed intensities. The results presented herein 9demonstrate that pulsating bleed flow is more efficient than its steady counterpart in terms of reducing pressure losses and controlling the primary frequency of the downstream oscillations for the same mass flow injection
Modifying such regions, i.e., including momentum source terms, flow stabilizers, or changing the shape of the body under study, we can control the appearance of flow instabilities. The method is tested in two different applications: the laminar flow behind a two-dimensional circular cylinder and a turbulent channel flow including heat transfer to the endwall. In the first test case, the flow unsteadiness is controlled by including two stabilizers (formed by two small cylinders) in the areas suggested by resolvent analysis. This is a benchmark problem in fluid dynamics that serves to validate the idea of using this tool for flow control. In the second test case, data-driven resolvent analysis is applied as a mean to control, enhance or reduce, convective heat transfer. Modifications in the geometry, in the form of cavities and ribs included in the areas pointed by resolvent analysis, show the possibility of enhancing heat transfer while reducing drag. The findings highlight the importance of resolvent analysis in understanding flow dynamics and designing effective flow control strategies.
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