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 of regularity, the filtering kernel must have a sufficient number of vanishing moments for the filtering spectrum to be meaningful. Our derivation follows a similar analysis by V. Perrier et al. [J. Math. Phys. 36, 1506 (1995)] for the wavelet spectrum, where we show that the filtering kernel has to have at least p vanishing moments 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, M-I and 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 nonquadratic 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 nonrectangular regions, in multiphase flows, or in geophysical flows on Earth's curved surface.