Extracting the Spectrum by Spatial Filtering

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.