Cold-boundary difficulty

The problem identified here is the one in which the cold-boundary difficulty, introduced in Section 2.2.2, first arose. It is clear that the trouble lies not in the mathematics but instead in the mathematical model of the physical situation the mathematical problem defined by equations (42) and (43) with the stated boundary conditions is ill-posed. A further indication of this fact may be obtained by differentiating equation (19) with respect to and using the result dejd = (de/dx)(dxldO = Am, derived from equations (42) and (19). We thereby see that... 

Avoidance of the appearance of the cold-boundary difficulty can be achieved only by revision of the physical model on which the mathematical formulation is based. There are many ways in which this can be done. In one approach, von Karman and Millan [17] replace t = 0 by t = (where 0 < < 1) as the position at which to apply the cold-boundary condition... 

With either of these methods of remedying the cold-boundary difficulty, the value of A depends, of course, on the chosen value of t,. It is found that the calculated estimate for A assumes a pseudostationary value as Tj, or the heat loss to the flame holder, is allowed to vary between reasonable limits [16]. This conclusion depends directly on the strong temperature dependence of the reaction-rate function. The pseudostationary behavior is illustrated in Figure 5.3, where the dashed line indicates the pseudostationary value of A. It is reasonable to identify the pseudostationary value as the... 

Other iterative methods have been proposed that are less well justified. For example, one approach [30] employs equations (48) and (49) directly, with the nth approximation for g(T) substituted into the integrals to obtain the (n + l)st approximations for A and g(z). A variational approach also has been developed [25], based on the introduction of / = as a new variable, so that equation (46) can be written in a standard form, d ildrj = — Aco/r], None of these approaches circumvent the cold-boundary difficulty unless is modified suitably, for example, by the introduction of 0. The approximations to be discussed next bypass the cold-boundary difficulty in a natural way. 

Let us first consider the relationship between e and i near the points defined by equations (8)-(ll). In the neighborhood of (i = 0, e = 0) [equations (8) and (9)], we must assume that F (t, (p) — 0 over a short range of i in order to remedy the cold-boundary difficulty (see Section 5.3.2). Hence equation (5-36) implies that e = 0 as i increases in the neighborhood of either of these points (that is, i > e here). 

The problem that has been defined here possesses the cold-boundary difficulty discussed in Section 5.3.2 and can be approached by the same variety of methods presented in Section 5.3. The asymptotic approach of Section 5.3.6 has been seen to be most attractive and will be adopted here. Thus we shall treat the Zel dovich number... 

It will be noted that these boundary conditions ( = 0 at t = and 1 at T = 1) are formally identical with the boundary conditions introduced in Section 5.3.2 in order to remedy the cold-boundary difficulty for the... 

Avoidance of the appearance of the cold-boundary difficulty can be achieved only by revision of the physical model on which the mathematical formulation is based. There are many ways in which this can be done. In one approach, von Karman and Millan [17] replace t = 0 by t = r, (where 0 < T < 1) as the position at which to apply the cold-boundary condition e = 0. The introduction of this artifice is equivalent to employing the physical concept of an ignition temperature I], below which the chemical reaction rate vanishes [see equations (19) and (33)]. An alternative procedure for determining a finite, nonzero value for A involves the assumption that a flame holder serves as a weak heat sink [18], removing an amount of heat per unit area per second equal to (XdT/dx)i = mCp(T - TQXdx/d )i. Since e = 0 at the flame holder (that is, there is no reaction upstream), equation (19) implies that dx/d i = rj, whence the heat-sink concept is seen to be mathematically equivalent to the ignition-temperature concept at the cold boundary. 

We have tested both discontinuous and smooth cutoff functions and find that the different possibilities have virtually no effect on the computed solution. The cutoff function is employed to avoid the "cold boundary difficulty" which arises because the Arrhenius model for the reaction term does not vanish far ahead of the front, which is incompatible with the boundary condition T = T as a —> 00. In addition, in practice no significant reaction occurs ahead of the reaction zone. For the computations presented here Qaut = 03. We have found that the results are insensitive to variations in Qcut as long as its value is of this order. 

---