Proper Orthogonal Decomposition Formulation Research Articles

Spectral proper orthogonal decomposition analysis of flowfield for supersonic nozzle jet noise reduction

Spectral proper orthogonal decompositions (SPOD) is the frequency domain form of proper orthogonal decomposition (POD). SPOD is shown to be directly applicable to jet noise studies. A computational SPOD analysis is performed on the Large-Eddy Simulation (LES) computational data for both the baseline nozzle flow and a modified nozzle flow. The mode energy analysis from SPOD shows that for the baseline nozzle, the low-rank behavior is found at frequencies at which a large separation occurs. The low-rank behavior disappears for the modified nozzle flow where no significant peaks are detected. The coherent structures of the low frequency show the Kelvin–Helmholtz type instability. It is also shown that the reconstructed flow field using the first decades of SPOD modes can capture the major shock-expansion based interactions that are present within the original flow field. This suggests that the method of SPOD-based reconstruction can be used to carry out additional modal and acoustic analyses of jet flows.

A conditional space–time POD formalism for intermittent and rare events: example of acoustic bursts in turbulent jets

We present a conditional space–time proper orthogonal decomposition (POD) formulation that is tailored to the eduction of the average, rare or intermittent events from an ensemble of realizations of a fluid process. By construction, the resulting spatio-temporal modes are coherent in space and over a predefined finite time horizon, and optimally capture the variance, or energy of the ensemble. For the example of intermittent acoustic radiation from a turbulent jet, we introduce a conditional expectation operator that focuses on the loudest events, as measured by a pressure probe in the far field and contained in the tail of the pressure signal’s probability distribution. Applied to high-fidelity simulation data, the method identifies a statistically significant ‘prototype’, or average acoustic burst event that is tracked over time. Most notably, the burst event can be traced back to its precursor, which opens up the possibility of prediction of an imminent burst. We furthermore investigate the mechanism underlying the prototypical burst event using linear stability theory and find that its structure and evolution are accurately predicted by optimal transient growth theory. The jet-noise problem demonstrates that the conditional space–time POD formulation applies even for systems with probability distributions that are not heavy-tailed, i.e. for systems in which events overlap and occur in rapid succession.

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.

Active Flow Control for High Speed Jets with Large Window PIV

The current work investigates a Mach 0.6 jet flow field with particle image velocimetry (PIV) and simultaneously sampled near and far-field pressure. Two component velocity measurements are taken in the streamwise plane of the jet. Three cameras are placed such that each interrogation window is captured simultaneously and stitched together to obtain a six diameter PIV window. In addition, active flow control is applied using an actuation glove comprised of synthetic jet actuators. Both open and closed-loop control are applied in different physical forcing configurations. For closed-loop control, hydrodynamic pressure from the near-field array of sensors is fed back to the actuation system in real time. The large window PIV allows one to examine how the flow field is affected by the flow control. Low-dimensional modeling techniques, in the form of proper orthogonal decomposition, are performed in order to obtain a better understanding of the large scale, energetic events in the flow field. It has been found that active flow control changes the potential core length and shear layer expansion, which affects the overall sound pressure levels in the far-field.

Structure identification in turbulent flows for feedback flow control

Feedback flow control strategies have traditionally been developed by focusing on prototype laminar flows such as the wake behind a cylinder or a free shear layer. Many strategies rely on some form of proper orthogonal decomposition (POD) of the velocity field to identify the flow structures that need to be mitigated. However, in turbulent flows, POD does not readily identify the large, coherent motion targeted by flow control. Ongoing research at the US Air Force Academy is focused on developing feedback flow control strategies for the turbulent shear layer behind a backward facing step. POD of the velocity and density fields showed that the density is a better marker of the coherent structures in the shear layer and nearly completely eliminates the contribution of the small-scale turbulent motion to the POD spatial modes, allowing for a focused development of strategies to control the coherent motion in the shear layer.

Reduced order computational methods for electromagnetic material interrogation using pulsed signals and conductive reflecting interfaces

We consider the interrogation by means of a pulsed planar electromagnetic wave of a dielectric slab with a supraconductive backing. Previous work using a weak formulation with finite elements (FE) demonstrated the ability to determine material parameters and the slab thickness in the inverse problem. In this work we report on results using Proper Orthogonal Decomposition (POD) to create a more efficient set of basis functions than the standard FE basis functions. We first demonstrate the ability of the reduced basis POD formulation to capture the electromagnetic behavior in the case of the forward problem. We then apply the POD formulation to the inverse scattering problem with unknown parameters and show that the POD formulation provides a considerable reduction in computational time over standard FE methods with comparable ability to recover the unknown parameter values.

Reduced order computational methods for electromagnetic material interrogation using pulsed signals and conductive reflecting interfaces

We consider the interrogation by means of a pulsed planar electromagnetic wave of a dielectric slab with a supraconductive backing. Previous work using a weak formulation with finite elements (FE) demonstrated the ability to determine material parameters and the slab thickness in the inverse problem. In this work we report on results using Proper Orthogonal Decomposition (POD) to create a more efficient set of basis functions than the standard FE basis functions. We first demonstrate the ability of the reduced basis POD formulation to capture the electromagnetic behavior in the case of the forward problem. We then apply the POD formulation to the inverse scattering problem with unknown parameters and show that the POD formulation provides a considerable reduction in computational time over standard FE methods with comparable ability to recover the unknown parameter values.