Three-dimensional wake division and scale recognition after a multi-scale fractal tree based on Hilbert–Huang transform

The aerodynamic performance of fractal geometries is a critical concern in engineering, such as urban trees, whose multi-scale wake structures warrant detailed examination. Identifying the contributions from multiple sub-scale tree geometries to wake dynamics requires decomposition and recognition of characteristic wake scales, which can encounter scale mixing in conventional methods such as the fast-fourier transform (FFT) and bare empirical mode decomposition (EMD). This study analyzes the wake characteristics behind fractal trees with varying crown porosities using the Hilbert–Huang transform (HHT). HHT decomposes the wake by EMD into intrinsic mode functions (IMFs) first, then determines each IMF's instantaneous frequency and amplitude by Hilbert spectral analysis. The statistical peaked frequency, which is calculated via the marginal Hilbert spectrum, reveals a similar major scale but a different spatial energy distribution compared with the FFT. The statistical joint probability density function (JPDF) of the instantaneous frequency and amplitude denotes consistent high-occurrence peaks at Sth = 0.2, 0.6, and 1.2, recommending including 1, 0.33, and 0.17 h into urban greening parameterization. The reconstructed peaked turbulence field exhibits specific spatial distribution patterns. This finding validates the bond between sub-scale geometries and JPDF peaks. The interaction of different peak scales is investigated to look into scale coherence. The central frequency and oval-like distribution slope offer a novel perspective on wake instability assessment. Additionally, the momentum fluxes driven by different peaked scales are examined, elucidating the contributions of sub-scale tree geometries to wake dynamics and pollutant transport, providing valuable guidance for optimizing urban greening design.