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, ${\mathcal{M}}^I$ 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 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.

Figure 1

Comparing Fourier and filtering spectra for a synthetic doubly periodic velocity field with a preset Fourier power-law spectrum (black): $E\left(K\right)\sim K^{-5/3}$ (upper left panel) and $E\left(K\right)\sim K^{-4}$ (upper right panel). The filtering spectra, $\overline{E}\left(K\right)$ are computed by convolving the velocity in $x$ space with top-hat (dark blue) and Gaussian (light blue) kernels. These two kernels correctly extract the $k^{-5/3}$ scaling, but not the $k^{-4}$, locking-in at a $k^{-3}$ scaling instead as described by Eq. (18). Lower two panels show the corresponding cumulative spectra, $\mathcal{E}\left(K\right)$, from which the filtering spectra are derived [Eqs. (14) and (15)]. Note that using any kernel, $\mathcal{E}\left(K\right)$ converges to the total energy (normalized to unity) at large $K$, demonstrating energy conservation [Eq. (20)].

Figure 2

Schematic of the SS kernels ${\mathcal{M}}^I\left(\mathbf{x}\right)$ (left panels) and ${\mathcal{M}}^I\left(\mathbf{x}\right)$ (right panels) we construct in 1D and 2D.

Figure 3

Energy spectrum for a doubly periodic synthetic velocity field with a preset power-law slope of $K^{-4}$. Figure compares the Fourier spectrum $(—)$ to the filtering spectrum using kernels ${\mathcal{M}}^I(—)$ and ${\mathcal{M}}^{II}(—)$, which we constructed above. Straight dashed black line with a $K^{-4}$ slope is for reference.

Figure 4

Demonstrating the “locking” effect implied by our result Eq. (18) when calculating the filtering spectrum. We use a doubly-periodic synthetic velocity field with a preset spectral slope of $k^{-7}$. We plot the filtering spectrum using top-hat $\left(—\right)$, ${\mathcal{M}}^I\left(—\right)$, and ${\mathcal{M}}^{II}\left(—\right)$ kernels, along with the Fourier spectrum $(—)$. We see that the filtering spectra using the three kernels (top-hat, ${\mathcal{M}}^I,{\mathcal{M}}^{II})$ lock at $k^{-3},k^{-5}$, and $k^{-7}$, respectively, consistent with Eq. (18).

Figure 5

Visualization of normalized vorticity (top-left) and kinetic energy (top-right) of the 2D periodic flow we analyze. The subdomain is highlighted as a dashed white box. To calculate the Fourier spectrum over the subdomain, the flow is periodized by tapering (lower-left) or mirroring (lower-right).

Figure 6

Figure of spectra from a doubly periodic 2D flow, normalized by the spatial average of energy over the entire domain. It shows the Fourier spectrum over the entire domain $(—)$ decaying slightly steeper than $k^{-3}$. It also shows the Fourier spectrum over the sub-domain highlighted in Fig. 5, calculated by the tapering $(––)$ and mirroring $(––)$ methods. We compare to the filtering spectra using a top-hat kernel $(—)$ and the ${\mathcal{M}}^{II}$ kernel $(—)$. Note that tapering, while it reduces the total energy in the subdomain, redistributes some energy from the largest scales to smaller scales, creating a significant bump over $k\in [2,20]$. The filtering spectrum with a top-hat kernel “locks” at a $k^{-3}$ scaling as expected for this flow. Both the filtering spectrum with the ${\mathcal{M}}^{II}$ kernel, and the Fourier spectrum using mirroring yield spectra that follow closely the Fourier spectrum over the entire domain.

Figure 7

A sample 2D slice of the normalized kinetic energy of a 3D flow from the JHTDB (left panel). The dashed white box bounds the subdomain of interest. Right panel shows the Fourier spectrum over the entire domain $(—)$. It also shows the Fourier spectrum over the subdomain periodized by the tapering $(––)$ and mirroring $(––)$ methods. We compare to the filtering spectra using a top-hat kernel $(—)$ and the ${\mathcal{M}}^{II}$ kernel $(—)$. All spectra are normalized by the spatial average of energy over the entire domain.

Figure 8

Streamlines of a large scale strain defined by $\mathbf{u}=(y,x)$ (top-left panel). We calculate the spectral content in the subdomain ${[-\pi /4,\pi /4]}^2$ (black dashed box). Top-right panel shows the filtering spectrum over the subdomain using ${\mathcal{M}}^{II}(—)$, and the Fourier spectrum by the mirroring $(--)$ and tapering $(--)$ methods. Inset plots the cumulative spectrum, $\mathcal{E}\left(k\right)$. Here, $k=L/\ell$, with $L=2\pi$. Bottom three panels show the kinetic energy of the unfiltered flow, ${\left|\mathbf{u}\right|}^2/2$ (left), and that of the filtered flow $|{\overline{\mathbf{u}}}_{\ell }{|}^2/2$ at $\ell =L/3$ (middle) and $\ell =L/6$ (right). All three bottom panels look almost identical, indicating the absence of small scales, consistent with the flow being a large-scale smooth coherent strain. This is revealed in the filtering spectrum, which shows almost zero spectral content at all small scales probed. In contrast, the Fourier spectrum is almost 12 orders of magnitude larger, exhibiting a power-law over a continuum of scales due to the spurious patterns that arise from mirroring. Both the Fourier spectrum and the filtering spectrum measure the same total energy as evidenced from $\mathcal{E}\left(k\right)$ at large $k$ in the inset of top-right panel.