Higher-order Spectral Moments Research Articles

Investigation of internal intermittency by way of higher-order spectral moments

The analysis of turbulence by way of higher-order spectral moments is uncommon, despite the relatively frequent use of such statistical analyses in other fields of physics and engineering. In this work, higher-order spectral moments are used to investigate the internal intermittency of the turbulent velocity and passive-scalar (temperature) fields. This study first introduces the theory behind higher-order spectral moments as they pertain to the field of turbulence. Then, a short-time Fourier-transform-based method is developed to estimate these higher-order spectral moments and provide a relative, scale-by-scale measure of intermittency. Experimental data are subsequently analysed and consist of measurements of homogeneous, isotropic, high-Reynolds-number, passive and active grid turbulence over the Reynolds-number range $35\leq R_{\lambda } \leq ~731$ . Emphasis is placed on third- and fourth-order spectral moments using the definitions formalised by Antoni (Mech. Syst. Signal Pr., vol. 20 (2), 2006, pp. 282–307), as such statistics are sensitive to transients and provide insight into deviations from Gaussian behaviour in grid turbulence. The higher-order spectral moments are also used to investigate the Reynolds (Péclet) number dependence of the internal intermittency of velocity and passive-scalar fields. The results demonstrate that the evolution of higher-order spectral moments with Reynolds number is strongly dependent on wavenumber. Finally, the relative levels of internal intermittency of the velocity and passive-scalar fields are compared and a higher level of internal intermittency in the inertial subrange of the scalar field is consistently observed, whereas a similar level of internal intermittency is observed for the velocity and passive-scalar fields for the high-Reynolds-number cases as the Kolmogorov length scale is approached.

Comparison and validation of physical wave parameterizations in spectral wave models

Recent developments in the physical parameterizations available in spectral wave models have already been validated, but there is little information on their relative performance especially with focus on the higher order spectral moments and wave partitions. This study concentrates on documenting their strengths and limitations using satellite measurements, buoy spectra, and a comparison between the different models. It is confirmed that all models perform well in terms of significant wave heights; however higher-order moments have larger errors. The partition wave quantities perform well in terms of direction and frequency but the magnitude and directional spread typically have larger discrepancies. The high-frequency tail is examined through the mean square slope using satellites and buoys. From this analysis it is clear that some models behave better than the others, suggesting their parameterizations match the physical processes reasonably well. However none of the models are entirely satisfactory, pointing to poorly constrained parameterizations or missing physical processes. The major space-time differences between the models are related to the swell field which stresses the importance of describing its evolution. An example swell field confirms the wave heights can be notably different between model configurations while the directional distributions remain similar. It is clear that all models have difficulty describing the directional spread. Therefore, knowledge of the source term directional distributions is paramount to improve the wave model physics in the future.

Analysis Tools for the Detection of IntermittentNonlinear Aeroelastic Phenomena

The nonlinear coupling between the aerodynamic load and structural motions in the experiments conducted in the Langley Transonic Dynamics Tunnel at NASA Langley Research Center on a flexible semispan model (FSM) of a high-speed civil transport wing configuration is examined. The results show that, right before hard flutter took place, the aerodynamic load and the structural motion of the FSM were intermittently and quadratically coupled. Particularly, the shock motion, which can be caused by local and intermittent flow separation, is nonlinearly related to the structural motion in such a manner that one full cycle of the shock motion is related to two cycles of the acceleration of the wing tip. Because of the required averaging in implementing them, Fourier-based higher-order spectral moments could not characterize this coupling. On the other hand, the wavelet-based cross bicoherence is established as a tool that is capable of detecting intermittent or transient nonlinear aeroelastic phenomena as would be encountered in model studies or flight tests.

Systematic search of bounds for total ?-electron energy

We developed a systematic method to improve the known bounds for the total π-electron energy E by designing a method based on the theory of symmetric polynomials and its inequalities. Additionally, the theory of symmetric polynomials applied to the problem of bounds for the total π-electron energy suggests the definition of generalized arithmetic and geometric means, which can then be incorporated into Kober's theorem to obtain very sharp upper and lower bounds, better than those already known, in terms of the determinant of the adjacency matrix of the graph and/or higher-order spectral moments. Having established a connection between the problem of the π-electron energy and the theory of symmetric polynomials and its inequalities, we apply several other theorems from the theory of symmetric polynomials that allow us to obtain new explicit and implicit inequalities and to obtain implicit inequalities that in principle incorporate all spectral moments. In particular, we present an implicit inequality for the total π-electron energy that is so sharp that it is for all practical purposes an equation for that energy. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Perspective: Measurements and Analyses of Nonlinear Wave Interactions With Higher-Order Spectral Moments

In fluid flows, fluid-structure interactions and other fluids-related problem, nonlinear dynamics play an important role in determining the development, response or output. Understanding these dynamics is essential for development of analytical models and prediction and control purposes. Higher-order statistical analysis has been shown to be an effective tool that can be applied to identify nonlinear couplings and measure energy transfer rates. These techniques have been applied by our group and others to investigate transition of shear flows, energy cascading in turbulence, oceanographic and geophysical flows, and fluid-structure interactions. The results of these investigations revealed important nonlinear characteristics regarding these problems. In this paper, we review these techniques and explain their usefulness in identification and quantification of nonlinear dynamics.

Defining the limitations of measurements from Doppler spectral recordings

The purpose of this study was to determine whether Doppler measurements of peak velocity and four other quantitative measures of spectral shape are affected significantly by the site of the Doppler recording in relation to the location of the maximum stenosis. Continuous-wave and pulsed Doppler recordings were made distal to a 70% (area reduction or 45% diameter reduction) asymmetric stenosis in an in vitro flow model under steady and pulsatile flow conditions. Recordings were taken at six different locations proximal and distal to the stenosis. A photochromic dye technique was used to visualize the actual flow field in the model. Distal to the stenosis, the flow visualization results demonstrated a strong radial and axial variation of the velocity field and thus explained why the Doppler measurements of peak frequency and spectral broadening were strongly dependent on the recording site. The peak frequency was maximum within the throat of the stenosis and returned to the prestenotic value five tube diameters distal to the stenosis. Other measurements of spectral broadening and spectral shape varied greatly depending on the location of the recording site in the poststenotic region. Higher order spectral moments such as the coefficient of kurtosis were found to exhibit large temporal variability, which makes them inappropriate as diagnostic indicators. Because of the complex nature of the poststenotic flow field, these results clearly demonstrate that no single Doppler measurement can accurately quantify the severity of a stenosis. Of the Doppler measurements only peak velocity is related to the severity of stenosis. Reproducible peak velocity measurements are obtained only if the Doppler sample volume is positioned at or very near the throat of the stenosis and at an appropriate radial site that may not necessarily be at the center of the vessel.

Application of higher order spectral analysis to cubically nonlinear system identification

In this study, digital higher-order spectral analysis and frequency-domain Volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time-invariant systems up to third order. The primary focus is on consideration of random excitation of nonlinear systems and, thus, the approach makes extensive use of higher-order spectral analysis to determine the frequency-domain Volterra kernels, which correspond to linear, quadratic, and cubic transfer functions. Although the Volterra model is nonlinear in terms of its input, it is linear in terms of its unknown transfer functions. Thus, a least squares approach is used to determine the optimal (in a least squares sense) set of linear, quadratic, and cubic transfer functions. Of particular practical note, is the fact that the approach of this paper is valid for non-Gaussian, as well as Gaussian, random excitation. It may also be utilized for multitone inputs. The complexity of the problem addressed in this paper arises from two principal causes: (1) the necessity to work in a 3D frequency space to describe cubically nonlinear systems, and (2) the necessity to characterize the non-Gaussian random excitation by computing higher-order spectral moments up to sixth order. A detailed description of the approach used to determine the nonlinear transfer functions, including considerations necessary for digital implementation, is presented. >

Quadratic system identification using higher order spectra of i.i.d. signals

The properties of higher order moment sequences and higher order spectral moments of an i.i.d. (independent, identically distributed) process up to fourth-order are discussed. These properties are utilized to develop algorithms to identify time-invariant nonlinear systems, which can be represented by second-order Volterra series and which are subjected to an i.i.d. input. A relatively simple solution for estimating the linear and quadratic transfer functions, which requires neither the calculation of the higher order spectral moments of the input for various frequencies nor the calculation of the inverse of matrix, is shown to exist, even though the second-order Volterra series is not an orthogonal model for an i.i.d. input (unless the input is a white Gaussian process). >

Fundamental–subharmonic interaction: effect of phase relation

The effect of the phase relation (i.e. phase difference and coupling) between the fundamental and subharmonic modes on the transition to turbulence of a mixing layer is investigated. Experiments are conducted to study the development of the subharmonic and fundamental modes under different phase-controlled excitations. Higher-order spectral moments are used to measure phase differences, levels of phase coupling, and energy transfer rates between the two modes at different downstream locations. Local measurements of the wavenumber–frequency spectra are used to examine the phase-speed matching conditions required for efficient energy transfer. The results show that when the phase coupling between the fundamental and the subharmonic is high, maximum subharmonic growth is found to occur at a critical phase difference close to zero. The subharmonic growth is found to result from a resonant parametric interaction between the fundamental and the subharmonic in which phase-speed matching conditions are satisfied. In contrast, when the phase coupling level is low, the phase difference is irregular and varying, the efficiency of parametric interactions is low, phase-speed matching conditions are not met and subharmonic growth is suppressed.

A non-Gaussian phenomenological model for higher-order spectra in turbulence

An extended Kolmogorov phenomenology for the dimensional analysis of the energy spectrum has been introduced by Kraichnan [Phys. Fluids 8 (1965) 1385]. Kraichnan's ideas are here extended to show how the higher-order spectral moments can scale as k − 5 3 , for any order and without the assumption of a Gaussian process. New experimental data on higher-order spectra is presented that exhibits this k - 5 3 scaling.

Higher-order spectral moments in collision-induced absorption. Inert gas mixtures

Expressions are derived for the two higher moments M(6) and M(8) of collision-induced absorption spectra of rare-gas mixtures. Numerical results are given for Xe-Ar at T = 295 K. The results will be useful in lineshape calculations.