POD based spectral Higher-Order Stochastic Estimation

Abstract

A new and unique analytical approach, capable of identifying both linear and higherorder coherences in multiple-I/O systems, is presented here in the context of turbulent flows. The technique is formed by combining two well-established methods: Proper Orthogonal Decomposition (POD) and Higher-Order Spectra Analysis; both of which were developed independently in the field of turbulence and system identification, respectively. The latter of these is based on known methods for characterizing nonlinear systems by way of Volterra functional series. In that, both linear and higher-order kernels are formed to quantify the nonlinear spectral transfer of energy between the system's input and output. This reduces essentially to spectral Linear Stochastic Estimation (LSE) when only the first-order terms are considered, and is therefore presented in the context of stochastic estimation as spectral Higher-Order Stochastic Estimation (HOSE). However, the trade-off to seeking higher-order transfer kernels is that the increased complexity restricts the analysis to single-I/O systems. Low-dimensional (POD based) analysis techniques are inserted to alleviate this void as the POD coefficients represent the dynamics of the so-called spatial structures (modes) of a multi degree-of-freedom system. At first, a Monte Carlo Simulation is performed to demonstrate the validity and characteristics of the higher-order Volterra series model. Next, the POD based spectral HOSE method is applied to an experimental data set comprising synchronous measures of the near-field pressure and far-field acoustics of a coaxial jet flow. Both near-field (line array of microphones) and far-field (arc array of microphones) signatures are decomposed independently using POD in order to obtain frequency dependent POD coefficients. The linear and quadratic kernels are computed using different combinations of POD coefficients. The results indicate that the acoustic signatures in the far-field of the jet are linearly related to the pressure signatures in the near-field pressure, as is expected in a subsonic Mach number jet flow.

Proper orthogonal decomposition-based spectral higher-order stochastic estimation

“…Therefore, the technique is being presented here in the context of stochastic estimation as spectral Higher-Order Stochastic Estimation (HOSE) but with the added benefit of including multiple points on the input and output sides of the system by way of POD. 28 A thirdorder Volterra approach was successfully applied by Park et al, 29 who considered the fluid-structure lock-in phenomena of a damaged fighter wing inducing unsteady flow. Linear, quadratic, and cubic coherence were obtained between the flow field (pressure) and structural response (acceleration) to characterize a nonlinear limit cycle oscillation of the wing.…”

Nonlinear Noise Propagation from a Fully Expanded Mach 3 Jet

“…By applying these techniques one can essentially identify the frequencies at which two signals are linearly and nonlinearly related. 25,23 Since the bispectral technique is an identification technique and does not provide the direct capability of estimation and prediction, another technique developed in the field of system identification was introduced which started with the work of Tick (1961 27 where it is combined with proper orthogonal decomposition to allow the application of this technique to sets of sensors in the near-field and far-field, thereby eliminating the limitation of single-sensor analysis. It is a promising technique but due to current practical limitations the technique is confined to second order (quadratic) systems and low frequency resolution.…”

Bispectral mode decomposition of nonlinear flows

“…To overcome these limitations in the context of stochastic estimation, Baars et al (2010) serially combined higher-order statistical analysis and modal decomposition by computing the cross-bispectrum between SPOD expansion coefficients from spatially separated, synchronized microphone arrays.…”