Space Time Proper Orthogonal Decomposition Research Articles

Time-resolved wave packet development in highly cooled hypersonic boundary layers

Boundary-layer disturbances are analysed on a $5^{\circ }$ half-angle blunted cone in Mach 5, high-enthalpy flow ( $h_0 = 9\ {\rm MJ}\ {\rm kg}^{-1}$ ) with a low wall-to-edge temperature ratio, $T_w/T_e = 0.18$ . Schlieren and focused laser differential interferometry (FLDI) are utilized to assess the structures and frequency content associated with disturbances. Wave packets are identified from bursts of modal content on time-resolved spectrograms. Bandpass filtering, proper orthogonal decomposition (POD) and space–time POD are then applied to the schlieren data. Bandpass filtering suggests the presence of wave packet dispersion and elongation indicative of slow-acoustic-wave synchronization. Modal reconstruction techniques indicate the radiation of content outside the boundary layer and distinct orientation changes within disturbances, potentially the first experimental evidence of the supersonic-mode instability in such a flow field. Cross-bicoherence computations are carried out for discrete time segments of data from both schlieren and FLDI data. They demonstrate that the most dominant nonlinear interactions are the fundamental–first-harmonic and the fundamental–low-frequency interactions.

Space-time POD and the Hankel matrix.

Time-delay embedding is an increasingly popular starting point for data-driven reduced-order modeling efforts. In particular, the singular value decomposition (SVD) of a block Hankel matrix formed from successive delay embeddings of the state of a dynamical system lies at the heart of several popular reduced-order modeling methods. In this paper, we show that the left singular vectors of this Hankel matrix are a discrete approximation of space-time proper orthogonal decomposition (POD) modes, and the singular values are square roots of the POD energies. Analogous to the connection between the SVD of a data matrix of snapshots and space-only POD, this connection establishes a clear interpretation of the Hankel modes grounded in classical theory, and we gain insights into the Hankel modes by instead analyzing the equivalent discrete space-time POD modes in terms of the correlation matrix formed by multiplying the Hankel matrix by its conjugate transpose. These insights include the distinct meaning of rows and columns, the implied norm in which the modes are optimal, the impact of the time step between snapshots on the modes, and an interpretation of the embedding dimension/height of the Hankel matrix in terms of the time window on which the modes are optimal. Moreover, the connections we establish offer opportunities to improve the convergence and computation time in certain practical cases, and to improve the accuracy of the modes with the same data. Finally, popular variants of POD, namely the standard space-only POD and spectral POD, are recovered in the limits that snapshots used to form each column of the Hankel matrix represent flow evolution over short and long times, respectively.

Proper Orthogonal Decomposition for a Turbulent Boundary Layer with Aero-Optic Distortion

The velocity field associated with the deflection of a laser beam passing through a heated incompressible boundary layer in the wall-normal direction was studied using variants of the classical proper orthogonal decomposition (POD) technique. Due to the heating of the boundary layer, temperature, density, and index of refraction fields were variable in the boundary layer and were instantaneously affected by turbulent coherent structures. The time-varying index of refraction field led to a time-varying beam path of a laser traversing the boundary layer; the final angle of the laser beam was measured simultaneously with particle image velocimetry. The relationship between coherent structures in the velocity field and the final beam angle was studied using conditional POD and space-time POD. Conditional POD was performed on the wall-normal velocity field using conditions of large upstream and downstream laser deflections. The modes from conditional POD were compared to modes from POD without conditions. Physical features within some conditional POD modes were found to be shifted in the streamwise direction compared to unconditioned modes, suggesting that the streamwise position of velocity features was correlated to laser deflection. Space-time POD resulted in the same modes as the conditional POD and showed the modes convecting in the streamwise direction without a significant change in the size of the structures. This suggested a hypothesis that the laser deflects when coherent structures reach particular positions relative to the laser position as the structures convect, aligning the largest local velocity and density gradients with the laser beam path.

Wind peak pressures on a square-section cylinder: Flow mechanism and standard/conditional POD analyses

This study emphasizes the flow mechanism of negative peak pressures on a square-section cylinder at Reynolds number of 22,000, particularly of those immediately upstream of the trailing edges (trailing-edge peak pressure). The minimum trailing-edge pressure coefficients are observed as low as -5 and have a close correlation in a streamwise length of approximately 0.1D (D is the width of the cylinder). The occurrence of small trailing-edge vortex, which has highly concentrated vorticity, is responsible for the trailing-edge peak pressures. The trailing-edge peak pressure occurs immediately before the shear layer moves far from the cylinder and recovers to the symmetric state. Standard proper orthogonal decomposition (POD) is applied to discover the inherent spatial modes with sharp peak pressures upstream of the trailing edges. Instead of the 1st mode (corresponding to vortex shedding), the higher-order modes contribute significantly to the trailing-edge peak pressures, whose mode coefficients have harmonics of vortex-shedding frequency. More than nine POD modes are necessary to reconstruct the trailing-edge peak pressures. As a variant of POD proposed recently, conditional space–time POD is valid in studying the average evolution of wind peak pressures. It further provides the relative phase and frequency ratio between the trailing-edge peak pressure and vortex shedding.

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.

Dynamic structures of a submerged jet interacting with a free surface

The spatial structures of a submerged jet interacting with a free surface were examined experimentally. The time-resolved PIV technique was used for the quantitative measurements. Two Reynolds numbers (1920, 3480) were examined for a configuration, in which the depth of the jet beneath the free surface was fixed to H=2D. The dynamics of the flow structures were examined further using the space–time proper orthogonal decomposition (POD) analysis technique. Some common characteristics were found in both jet flows, which are believed to be the general features of the free-surface jet flows. On the other hand, the dynamic characteristics of the large-scale turbulent motions differed significantly with respect to the Reynolds number. In the low Reynolds number case, three dominant dynamic structures were found: flapping motion of the jet, very large scale vortical structure interacting with the free surface at the right side, and the transverse stretching of large scale structure due to the presence of free surface. The reconstructed fluctuating velocity fields and the swirling strength contours demonstrated that the very large scale structures began to move downward at about 10D downstream the initial interaction. In the high Reynolds number case, the first two dominant spatial modes showed the momentum exchange process between the jet flow and the free surface. The third mode displayed a vortex just below the free surface at the left side and the fourth mode showed the elongated vortex structures. The scale of vortical structures was much smaller in the high Reynolds number case than that in the low Reynolds case, and the downward motion of large scale structures was not found in the high Reynolds number case.