Scaling analysis for buoyant plumes over wildland fires

TL;DR

The paper develops a dimensional-analysis framework using Buckingham's theorem to identify dimensionless groups ($\pi$) that govern buoyant plume trajectories in cross-wind, with and without vegetative canopies. It validates these scaling relations against classical cross-wind plume theory and tests them via large-eddy simulations that span canopy and no-canopy conditions, various heat-source strengths, and ambient wind speeds. Key findings show that canopy presence markedly increases near-surface plume slopes (about an order of magnitude) due to canopy drag and delays the rise-to-bent-over transition, while far-field slopes remain sensitive to the same buoyancy-to-wind ratio but with canopy-adjusted dynamics. The results yield practical scaling laws linking plume centerline slope and transition height to the dimensionless groups, offering improved predictive capability for smoke transport and fire-management models across diverse environments.

Abstract

Tracking the structure and geometric properties of a buoyant plume in cross-wind is critical for managing smoke hazards and improving disaster mitigation efforts. Plume features, such as the tilt angle, centerline trajectory, plume height, and curvature changes with height, are impacted by a range of forcing parameters, with the altered turbulence patterns induced by a vegetative canopy introducing an added layer of complexity. This study examines the effects of these parameters, reduced to a set of fewer dimensionless groups, on the plume centerline slope both, near the surface and in the far-field (bent-over phase). Results from a suite of large-eddy simulations in both canopy and no-canopy environments explore power-law dependencies between the slopes and key dimensionless groups describing (1) the relative strength of the buoyancy source to the ambient wind forcing and (2) the turbulence intensity within the plume relative to upstream. Near-surface slopes are an order of magnitude higher in the canopy cases owing to canopy drag. In the canopy cases, the near-surface plume slope increases markedly with increase in the dimensionless plume turbulence intensity, exhibiting a one-fourth power-law dependence. This effect is absent in the no-canopy case, reflecting spatial differences in the momentum-flux structure near the plume source between the two environments. Moreover, the canopy aerodynamic effects delay the transition of the plume from the rise phase into the far-field compared to the no-canopy cases. The transition height follows a one-fourth and one-third power-law dependence on group (1) in the canopy and no-canopy environments, respectively, with canopy effects becoming less prominent at higher buoyancy source strength. Our findings support the development of scaling laws for plume structures across varied environments and inform improved predictive modeling.

Figures (5)

• Figure 1: Schematic 2-D representation of the plume interacting with a cross-wind based on a linear plume rise model. After an initial rise phase, plume centerline trajectory transitions to the bent-over phase (sometimes called the entrainment phase) at $(x_\text{pc},~z_\text{pc})$, where the centerline undergoes a change in slope
• Figure 2: Contours of the 1-h mean vertical velocity ($\overline{w}$) for the (a) no-canopy and (b) canopy environments depicting the respective plume centerline inclinations in each zone. Trend-lines in Zone 2 are obtained using MATLAB's fminsearch() for a linear fit to the oscillating mean plume centerline as done by A. Desai, A. Q. Cervantes, T. Banerjee (manuscript in preparation)
• Figure 3: Plot of plume centerline slope ($\Delta\overline{z}/\Delta x$) against (a) $\pi_3\pi_4$ and (b) $u_*^{(\text{pl})}/u_*^{(\text{up})}$ in Zones 1, 1(B), and 2 for the canopy cases. Slope i, computed from the simulation outputs, refers to the plume centerline slope in Zone i. Trend-lines represent the best fit of a power law to the data; $R^2$ represents the coefficient of determination
• Figure 4: Plot of plume centerline slope ($\Delta\overline{z}/\Delta x$) against (a) $\pi_3\pi_4$ and (b) $u_*^{(\text{pl})}/u_*^{(\text{up})}$ in Zones 1--2 for the no-canopy cases. Slope i, computed from the simulation outputs, refers to the plume centerline slope in Zone i. Trend-lines represent the best fit of a power law to the data; $R^2$ represents the coefficient of determination
• Figure 5: Plot comparing the normalized height ($z_\text{pc}/\delta$) at which the plume transitions from the rise phase to the bent-over phase (Zone 2) between the canopy and no-canopy cases. Trend-lines for $z_\text{pc}/\delta$ represent power-law dependence; $L_\text{B}/\delta$ represents the normalized buoyancy length scale