Extracting the spectrum of a flow 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 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 [1] 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) less/equal to p + 2. For example, any ow 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, MI and MII, 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 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 or in geophysical flows on Earth's curved surface.