A simple and fast numerical method for solving flame/smoldering evolution equations

Abstract

We propose a simple and fast numerical method for solving an evolution equation for closed flame/smoldering fronts, equivalent to the Kuramoto-Sivashinsky equation in a scale. Comparison of numerical results and an experiment suggests that our model equation is valid for not only propagating gas-phase flame fronts but also expanding smoldering fronts over thin solids.

The existence of intrinsic rotating wave solutions of a flame/smoldering-front evolution equation

“…Goto et al [5] proposed a fast and simple numerical method for front tracking of (2) and reported that (2) is valid not only for propagating gaseous-flame fronts, but also for expanding smoldering fronts over thin solids.…”

A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies

“…The purpose of this section is to define in a discrete sense the "curvature" κ and its second "derivative" κ ss arising in (6) and (7). We use a similar way of space discretization and computation of the tangential velocity to [4]. In the direct approach, a moving Jordan curve Γ(t) is approximated by a moving Jordan polygonal curve, say P(t) at time t, with N vertices labeled…”

“…and ω is a sufficiently large constant. Then, r i → L/N (t → ∞) is satisfied (see [4,8,9,11] in detail). From the above, (10) can be summarized as the following ODEs:…”

“…Moving boundary problems have attracted many researchers and been extensively studied in the fields of not only applied mathematics but also physics such as combustion and fire research. In recent studies, spreading flame/smoldering fronts along a sheet of paper were tracked experimentally (see Figure 1 and [5]), and numerically (see Figure 2 and [4]) by applying the Kuramoto-Sivashinsky (KS in short) equation [3,7,12]. The purpose of this paper is to detect the fronts from two dimensional experimental images, and to determine the "real" values of the parameters that appear in the KS equation.…”

“…Therefore, the shape is thoroughly determined by the normal velocity V only and we can choose the value of W according to our convenience. The following equation is a version of the KS equation for gas combustion [3], which is also a model equation for flame/smoldering fronts along a sheet of paper [4,5]:…”