Drag Crisis in Fractal Trees Revealed by Simulation and Theory

Abstract

Trees are key roughness elements in urban environments, shaping airflow, microclimates, and pollutant dispersion. Yet the aerodynamic drag of complex tree-like structures at high Reynolds numbers remains poorly characterized compared with the well-studied drag crisis of simple bluff bodies. We combine large-scale lattice Boltzmann simulations with an analytical branch-wise drag model to examine fractal trees over a wide range of height-based Reynolds numbers, $Re_H$. Direct numerical simulations using a cumulant lattice Boltzmann method with adaptive mesh refinement cover $2.5\times10^3 \le Re_H \le 1.2\times10^5$, and the analytical model extends predictions to $Re_H \sim 10^9$. Under uniform inflow, the analysis indicates a drag-crisis transition near $Re_H \approx 3\times10^6$, with increasing structural complexity smoothing this transition because smaller branches remain subcritical. Introducing inflow turbulence with streamwise intensity $I_u \approx 8\%$, representative of atmospheric-boundary-layer winds, shifts the apparent onset to $Re_H \approx 1.5\times10^5$ and further moderates the drag reduction. Interpreted at full scale, this suggests that urban trees of order $10$--$30$ m exposed to winds of $1$--$10~\mathrm{m/s}$ generally operate in the crisis or post-crisis regime. In both uniform and turbulent inflow, the framework predicts a reversal in drag-coefficient ordering across geometries: simplified trees show lower drag in the subcritical regime but may exhibit higher drag in the supercritical regime, whereas more complex trees undergo a smoother, moderated crisis. These results challenge the common assumption that pruning always reduces aerodynamic loading and highlight the need to reassess vegetation-drag parameterizations and pruning strategies in high-$Re_H$ conditions.

Figures (8)

• Figure 1: Axis conventions $(+, -, \&, \wedge, \backslash, /)$ for the L-system turtle interpretation. The vectors $\vec{H}$, $\vec{L}$, and $\vec{U}$ denote the heading, left, and up directions, respectively.
• Figure 2: Fractal tree geometries. Top row: side views for (a) $n=4$, (b) $n=6$, and (c) $n=8$. Bottom row: corresponding top views for (d) $n=4$, (e) $n=6$, and (f) $n=8$. The parameter $n$ denotes the number of recursive branching generations.
• Figure 3: Computational domain and arrangement of trees. $U_\infty$ represents the uniform flow velocity in the $X$-direction of the inflow condition. The $(X,Y,Z)$ coordinate system is used for the numerical simulations.
• Figure 4: Hierarchical computational mesh around the Basic $n=8$ tree model. The mesh is refined progressively from the far field to the tree and wake regions. The finest resolution, $\Delta x/H = 1/2048$, is applied near the smallest branches, while coarser resolutions are used farther away and in the wake region ($\Delta x/H = 1/256$).
• Figure 5: Schematic of the analytical framework. The tree-like structure is decomposed into a series of individual cylindrical segments. The orientation of each segment is defined by the angle $\alpha$ between its central axis and the streamwise direction of the uniform inflow velocity.