Flame-sheet example

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

All the assumptions of Section 1.3 underlie the formulation. Assumption 1 deserves special mention because buoyancy forces on the hotter, less dense gas in the region of the flame often distort the shape of the flame sheet. A Froude number, Fr = v l(ag) (where g denotes the acceleration of gravity), measures the relative importance of inertial and buoyant forces. If Fr is sufficiently small (for example, Fr Ap/p, where Ap is a representative density difference and p a mean density), then the flame heights are controlled by buoyancy. Correlations of measured flame heights in the form hja Fr , where h is the flame height and < n are available for b 00 under buoyancy-influenced conditions with negligible viscous forces [9]-[ll]. The result of equation (26), namely, hja vajD = Pe, is a Peclet-number (Pe) dependence that cannot be correlated with Fr and that... 

It may be observed that only the fuel and oxidizer concentration fields have been considered in finding the flame shape. The nature of the boundary conditions makes it unnecessary to study the temperature- and product-concentration fields when the stated assumptions are adopted. If temperature or product concentrations are desired, they may be calculated a posteriori, in terms of the known fuel or oxidizer fields, by solving equation (1-49) for and Pi with = otp, for example. Temperatures at the flame sheet calculated in this way usually are too high (see Section 3.4). 

Since equation (42) is valid for any coupling function other results can be derived without explicitly invoking the flame-sheet approximation. For example, by considering... 

The mixture fraction Z could have been introduced and employed directly in earlier examples. Since it is a coupling function, it could have been used in place of P in equations (9) and (38), with equation (70) employed to recover jS from the solution for Z. In fact, it could have been introduced in Section 1.3, to replace P in equation (1-49). Although such selections of variables basically are matters of personal taste, the replacement of jS by Z achieves a convenient normalization and also can help to clarify aspects of physical interpretations. For example, in equation (25) the flame-sheet condition, jS = 0, becomes Z = Z, a condition of mixture-fraction stoichiometry. For the droplet-burning problem, when all the assumptions that underlie equation (58) are introduced, it is found that equation (42), interpreted for Z, becomes simply Z = 1 — (1 + where B is defined at... 

The boundary conditions that will be adopted at y = 0 (where conditions will be identified by the subscript 0) are = 0,(Xp = (Xp q = constant (a measure of the sublimation or boiling temperature), and u = 0. The first of these conditions would follow from a flame-sheet hypothesis and is accurate for many liquid fuels (see Section 3.3.4 or [21]). The validity of the second condition is discussed in Section 3.3.4 this condition will be most accurate for a volatile liquid fuel. The third condition is rigorously true only for a solid fuel but is an excellent approximation for liquid fuels if longitudinal flow of the liquid is surpressed (for example, by extruding the liquid through a porous solid material). ... 

With F(rj) known, we may readily calculate u from equation (28) and pj from equation (47). Obtaining profiles in physical coordinates involves first evaluating p(/ )—for example, from pj and the ideal-gas law by introduction of a flame-sheet approximation—and then performing the integration in equation (27) for y. Profiles of the normal mass flux, pv, are given by equation (29), which—for the present problem—reduces to... 

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