On a class of SGS filters for LES with shape preserving concatenation properties
The problem throughout is that the developed filter functions are based on a pure 1D analysis, although LES filtering is a true 3D-process (for 3D turbulence). All integrals presented therein are therefore not 1D but have to be identified as 3D integrals, the coordinates $x$ and $k$, as well as the coordinate differences $x-y$, have to be identified as vector norms $|\mathbf{x}|$, $|\mathbf{k}|$, $|\mathbf{x}-\mathbf{y}|$, and usual products between coordinates as scalar products. Thus Eq.(7) has to be evaluated as a 3D Fourier-transform and not as a 1D Fourier-transform, as incorrectly done throughout this study. This will lead to different explicit results for the filter functions, even if they are assumed to be spherically symmetric — although this assumption of spherically symmetric filter kernels is not very feasible when it comes to anisotropic flow fields.
Nevertheless, the key idea of this study, namely to construct filter kernels that have shape preserving concatenation properties when successively applying them onto a flow field for different filter widths, remains valid. The author was lucky that the key equation, Eq.(9), to derive these properties, remains intact in 3D under a spherically symmetric ansatz: Up to a different normalization constant, the 1D variable $k$ in Eq.(9) just has to be identified as the 3D vector norm $|\mathbf{k}|$.
Two more remarks: The left-hand side of Eq.(19) has a typo: instead of $\Phi(t)$ it should be $\Gamma(t)$. And then a fun fact: Coarse graining turbulent fluxes with a spherically symmetric filter kernel guarantees their invariance under the full Galilean symmetry group in non-relativistic flows, but in relativistic flows there exists no possible space-time coarse graining that can preserve Lorentz symmetry (see G.L. Eyink & T.D. Drivas 2018, Cascades and dissipative anomalies in relativistic fluid turbulence, https://doi.org/10.1103/PhysRevX.8.011023).