Elliptical fire front speed function R. McDermott∗ and R. Rehm Building and Fire Research Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899-8663, USA

Abstract

It is often postulated that a wind-driven fire front emanating from a point source ignition will evolve into an elliptical shape with the long axis of the ellipse aligned with the wind direction. The leading edge or head of the front propagates fastest, the trailing edge or tail propagates upwind at a fraction of the head speed, and the side edges or flanks are retarded compared to the head. A front tracking speed function is needed to model the fire front by a level set method, for example. Often when modeling fire front propagation the speed function is stated a priori, leaving little control over the shape of the resulting surface. Here we start from the assumption that a point source ignition will evolve into an ellipse and from there work backwards to discover the form required by the speed function used within a level set method.

1

Introduction

Previous authors [1, 2, 3] have used the level set approach of Sethian [4] to describe the evolution of a fire front with and without wind forcing. Consider the fire front to be given by the zero level set of the scalar φ(x, y, t), where x and y are 2D spatial coordinates and t is time. The scalar field evolves by the level set equation ∂φ + F · ∇φ = 0 , ∂t ∗

(1)

Corresponding author. Email: [email protected]

NIST Technical Note Preprint

15 September 2008

Figure 1: Elliptical fire front. The initial front is a circle of radius r centered below the origin o. A short time later the front has evolved into an ellipse centered at the origin with short semi-axis a and long semi-axis b.

where F is the speed function of the fire front. The operator on the left hand side of (1) is analogous to the material derivative D/Dt = ∂/∂t + U · ∇ where now F plays the role of the fluid velocity. Thus an element following the “streamline” of F which is initially zero will remain so. In this work we imagine that the initial fire front is a circle of radius r located at (0, −c ∆t v) where v is the wind speed in the positive y direction, ∆t is a small time increment, and 0 < c < 1 is a free parameter relating the relative speed of the fire front to the ambient wind (see Figure 1). We prescribe that at a short time ∆t later the fire front evolves into an ellipse with parameters a and b centered at the origin o. The speed function implicit in such a transformation is F = f0 ξ +

bc vs − cvn , r

(2)

where f0 is the front speed without wind forcing, ξ = ∇φ/|∇φ| is the unit normal vector for the fire front surface, s is the unit vector in the wind direction, and n is the unit vector normal to the wind direction (the sign on n will be discussed later). We note that the general form of (2) is not new, but our derivation given below provides insight which may be helpful in isolating model parameters in experiments and model 2

validation exercises.

2

Derivation

2.1

Wind-driven front propagation at short times

The position of an element of the fire front is given by (X[t], Y [t]). The position of the center of the circular front at the initial time is (0, −c ∆t v). With the assumptions given above, i.e., that the ellipse is centered at the origin and that the wind has magnitude v in the plus y direction, we may write the equations for the circle as X[t]2 + (Y [t] + c ∆t v)2 = 1, r2

(3)

X[t + ∆t]2 Y [t + ∆t]2 (X[t] + ∆tFx )2 (Y [t] + ∆tFy )2 + = + = 1. a2 b2 a2 b2

(4)

and for the ellipse as

Equating (3) and (4), and examining the solution for X(t) = 0 (implies Y (t) = r and Fx = 0 by symmetry) we find

Solving for Fy we have

(r + ∆tFy )2 (r + c ∆t v)2 − = 0. b2 r2

(5)

b(1 + rc ∆t v) − r bc Fy = lim = v. ∆t→0 ∆t r

(6)

Again, equating (3) and (4), now taking Y (t) = 0, we have X(t) = r, but note that Fx 6= 0. We then obtain (r + ∆tFx )2 (∆tFy )2 (c ∆t v)2 − 1 + − = 0. a2 b2 r2

(7)

Using (6) we see that the last two terms on the left hand side of (7) cancel and we are left with a−r . ∆t→0 ∆t

Fx = lim

3

(8)

The parameter a may be eliminated by noting that a → b as (∆t v) → 0. Thus, at short times we may take b = a + c ∆t v. The x component of the speed function may now be determined as b − r − c ∆t v = −cv . ∆t→0 ∆t

Fx = lim

2.2

(9)

In the absence of wind

The previous section considers the change in shape of the fire front for very short times in the presence of a mean wind. Here we specify that in the absence of a mean wind the fire front will propagate uniformly in all directions maintaining a circular shape. With v = 0, the resulting speed function is F = f0 ξ ,

(10)

where f0 is the front propagation speed in the absence of wind and ξ = ∇φ/|∇φ| is the unit normal vector to the fire front surface pointing from the burned to the unburned area.

2.3

Rotation of the wind direction

So far the derivation has assumed that the wind is constant and uniform and blowing in the positive y direction. However, in the more general case we wish to apply our speed function to a vector wind field v(x, t) which varies spatially and temporally. Let v ≡ |v| and let s = v/|v| denote the unit vector in the direction of v. Additionally, let n denote the unit vector which is orthogonal to s (sign to be determined). That is, n = ± [sy , −sx ]T .

(11)

We now have discussed each of the components in the formula (2). As mentioned before, the last remaining question is the sign of n. For the simple case used in the derivation, where the wind propagates in the positive y direction, we have s = [0 v]T . Therefore, on the right side of the ellipse, i.e., where ξx > 0, we require the term −cv[±nx ] = −cv[±sy ] = −cv[±v] < 0 in order to retard the motion of the fire front along the flank. This tells us that we should choose the positive sign for the case when the surface normal is pointing to the 4

Figure 2: Rotated fire front.

right of the wind direction. Stated another way, the wind normal points from the long axis of the ellipse toward the ellipse surface (see Figure 2). For a spatially varying wind blowing in an arbitrary direction θ = arcsin(sx ) relative to due north (again, see Figure 2), we may determine the sign on the flank unit vector n by first rotating the fire front surface normal ξ by θ to obtain ξ ′ and then taking the sign of n as the sign of ξx′ . That is, ξx′ = ξx cos θ − ξy sin θ , ξy′ = ξy cos θ + ξx sin θ , and

8 > < +[sy , n=> : −[sy ,

−sx ]T if ξx′ > 0 −sx ]T if ξx′ < 0 .

5

(12)

(13)

3

Results

In this section we present results for two cases corresponding to θ = [0, π/4] (see Figure 3) using the following model parameters: v

= 1.0

f0

= 1.0

b/r = 4.0 c

= 0.2

We have already mentioned that the parameter c represents the speed of the wind-driven fire front relative to the wind speed. The fire front typically propagates at some small (but not vanishingly small) fraction of the wind speed. It is safe to say that 0 < c < 1. To prevent the front from propagating into a burned area and for “back burn” to occur (i.e., for the fire to propagate upwind), we must have bc/r 6 f0 . Additionally, an implicit assumption in the elliptical model is that b/r > 1. We approach a point source ignition as r → 0.

Figure 3: Level set results for the speed function, Eq. (2), for the case θ = 0 (left, wind is from the south) and case θ = π/4 (right, wind is from the southwest).

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4

Conclusions

In this work we present a simple derivation of a fire front speed function starting from the generally accepted premise that a fire front emanating from a point ignition source will evolve into elliptical shape in the presence of a wind. The general concept is not new, but our derivation is insightful in that each of the model parameters has a clearly defined physical interpretation that may be helpful in designing experiments to validate front propagation models. Further, the general form of the speed function is not limited to point ignition sources: line sources of arbitrary shape may be applied to initiate the front and the same general numerical procedure used to generate the results presented here can be used.

References [1] J. L. Coen. Applications of coupled atmosphere-fire modeling: Prototype demonstration of realtime modeling of fire behavior. In Joint 6th Symposium on Fire and Forest Meteorology/Interior West Fire Council Conference, 2005. [2] F. E. Fendell and M. F. Wolff. Wind-Aided Fire Spread, Forest Fires, Behavior and Ecological Effects, chapter 6, pages 171–223. 2001. [3] V. Mallet, D. E. Keyes, and F. E. Fendell. Modeling wildland fire propagation with level set methods, 2007. [4] J. A. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge, second edition, 1999.

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