Modelling fire line merging using plane curvature flow
Piantadosi, J., Anderssen, R.S. and Boland J. (eds) MODSIM2013, 20th International Congress on Modelling and Simulation · 2013
The merging of two lines of fire is a relatively common occurrence in landscape fire events.For example, it can arise through the coalescence of two wildfires or when a prescribed fire meets a wildfire as part of suppression efforts.When two fires approach one another, the effects of convective and radiative heat transfer are compounded and high rates of spread can arise as a result.This is particularly the case when two oblique lines of fire meet at some acute angle -the point of intersection on the newly merged fire can advance rapidly.This case was investigated recently by Viegas et al. (2012), who devised a simple analytical model to emulate the effects of energy concentration between the two merging fire lines.In this paper, we present a more geometric approach by considering the evolution of the merged fire as the flow of a plane curve with a normal speed that depends on the curvature of the fire front.Specifically, we formulate the curvature flow equations in terms of a time-varying graph y(x, t).The resulting evolution equation is a quasilinear degenerate parabolic second-order partial differential equation:The evolution of the merged fire lines is then modelled by solving an associated initial value and boundary value problem, where the initial conditions are taken y 0 (x) ≈ |x| tan θ.The θ here controls the initial angle between the two merged fire lines.Parametric variation of the curvature dependence (via the parameter ǫ) is investigated, and the resultant geometric evolutions of the fire front are compared with the experimental observations of Viegas et al. (2012).The curvature flow simulations were able to capture a number of features that were observed by Viegas et al. (2012).In particular, the model was able to reproduce the 'rotation' of the two fire lines noted by Viegas et al. (2012) and was able to account for the qualititative rate of spread behaviour observed in connection with the rapid advance of the point of intersection of the merged fire lines.This so-called jump velocity profile was characterised by very high initial rates of spread, immediately after the fire lines merged, and then a gradual slowing of the rate of spread approaching a quasi-steady value.Further theoretical aspects of plane curvature flows and their more general application to fire front modelling are discussed and a number of improvements to the model are suggested.