Extracting the Spectrum by Spatial Filtering

TL;DR

This paper introduces a local physical-space filtering method to extract flow spectra, demonstrating its effectiveness and advantages over wavelet analysis in various turbulent flow scenarios.

Contribution

The authors develop simple stencil kernels with sufficient vanishing moments to extract spectra steeper than $k^{-3}$, enabling local spectral analysis in complex flows.

Findings

Method accurately extracts spectra from synthetic and turbulent flow data.

Constructed kernels can handle spectra steeper than $k^{-3}$.

Guarantees energy conservation and handles non-quadratic quantities.

Abstract

We show that the spectrum of a flow field can be extracted within a local region by straightforward filtering in physical space. We find that for a flow with a certain level of regularity, the filtering kernel must have a sufficient number of vanishing moments in order for the "filtering spectrum" to be meaningful. Our derivation follows a similar analysis by Perrier et al. 1995 for the wavelet spectrum, where we show that the filtering kernel has to have at least $p$ vanishing moments in order to correctly extract a spectrum $k^{-\alpha}$ with $\alpha < p+2$. For example, any flow with a spectrum shallower than $k^{-3}$ can be extracted by a straightforward average on grid-cells of a stencil. We construct two new "simple stencil" kernels, ${\mathcal M}^{I}$ and ${\mathcal M}^{II}$, with only two and three fixed stencil weight coefficients, respectively, and that have sufficient vanishing moments to allow for extracting spectra steeper than $k^{-3}$. We demonstrate our results using synthetic fields, 2D turbulence from a Direct Numerical Simulation, and 3D turbulence from the JHU Database. Our method guarantees energy conservation and can extract spectra of non-quadratic quantities self-consistently, such as kinetic energy in variable density flows, which the wavelet spectrum cannot. The method can be useful in both simulations and experiments when a straightforward Fourier analysis is not justified, such as within coherent flow structures covering non-rectangular regions, in multi-phase flows, or in geophysical flows on Earth's curved surface.