Mixture-fraction PDF

In a turbulent flow for which it is possible to define the mixture-fraction vector, turbulent mixing can be described by the joint one-point mixture-fraction PDF MC x, t). The mean mixture-fraction vector and covariance matrix are defined, respectively, by76 i 

Since the mixture-fraction PDF is not known a priori, it must be modeled either by solving an appropriate transport equation77 or by assuming a functional form. 

The most widely used approach for approximating / ( x, t) is the presumed PDF method, in which a known distribution function is chosen to represent the mixture-fraction PDF. We will look at the various possible forms in Section 5.9, where presumed PDF 

76 By convention, the mixture-fraction PDF is defined to be null whenever the sum of the components J, is greater than unity. Thus, the upper limit of the integrals in these definitions is set to unity. 

The beta PDF contains two parameters that are functions of space and time a(x, t) and b(, t). The normalization factor B(a, b) is the beta function, and can be expressed in terms of factorials  

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Moreover, the joint composition, mixture-fraction PDF can be written as... 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

The interest in reformulating the conserved-variable scalars in terms of the mixture-fraction vector lies in the fact that relatively simple forms for the mixture-fraction PDF can be employed to describe the reacting scalars. However, if < /Vmf, then the incentive is greatly diminished since more mixture-fraction-component transport equations (Nmf) would have to be solved than conserved-variable-scalar transport equations (/V, << ). We will thus assume that N m = Nmf and seek to define the mixture-fraction vector only for this case. Nonetheless, in order for the mixture-fraction PDF method to be applicable to the reacting scalars, they must form a linear mixture defined in terms of the components of the mixture-fraction vector. In some cases, the existence of linear mixtures is evident from the initial/inlet conditions however, this need not always be the case. Thus, in this section, a general method for defining the mixture-fraction vector in terms of a linear-mixture basis for arbitrary initial/inlet conditions is developed. 

The determination of a mixture-fraction basis is a necessary but not a sufficient condition for using the mixture-fraction PDF method to treat a turbulent reacting flow in the fast-chemistry limit. In order to understand why this is so, note that the mixture-fraction basis is defined in terms of the conserved-variable scalars pcv without regard to the reacting scalars pT. Thus, it is possible that a mixture-fraction basis can be found for the conserved-variable scalars that does not apply to the At reacting scalars. In order to ensure that this is not the case, the linear transformation Mr defined by (5.30) on p. 149 must be applied to the (K x VIM ) matrix... 

Figure 5.8. The mixture-fraction PDF in turbulent flows with two feed streams (binary mixing) can be approximated by a beta PDF. 

The beta PDF is widely used in commercial CFD codes to approximate the mixture-fraction PDF for binary mixing. This choice is motivated by the fact that in many of the canonical turbulent mixing configurations (Fig. 5.8) the experimentally observed mixture-fraction PDF is well approximated by a beta PDF. However, it is important to note that all of these flows are stationary with Nmf = A m — 1 = 1, i.e., no linear mixture exists between the inlet conditions. The unmixed PDF is thus well represented by two peaks one located at % = 0 and the other at % = 1, which is exactly the type of behavior exhibited... 

Figure 5.9. The unmixed mixture-fraction PDF in turbulent flows with two feed streams has two peaks that can be approximated by a beta PDF. However, with three feed streams, the unmixed PDF has three peaks, and is therefore poorly approximated by a beta PDF. 

When the mixture-fraction vector has more than one component, the presumed form for the mixture-fraction PDF must be defined such that it will be non-zero only when AW... 

Chapter 3 will be employed. Thus, in lieu of (x, t), only the mixture-fraction means ( ) and covariances ( , F) (/, j e 1,..., Nm() will be available. Given this information, we would then like to compute the reacting-scalar means and covariances (require additional information about the mixture-fraction PDF. A similar problem arises when a large-eddy simulation (LES) of the mixture-fraction vector is employed. In this case, the resolved-scale mixture-fraction vector (x, t) is known, but the sub-grid-scale (SGS) fluctuations are not resolved. Instead, a transport equation for the SGS mixture-fraction covariance can be solved, but information about the SGS mixture-fraction PDF is still required to compute the resolved-scale reacting-scalar fields. 

In a CFD calculation, one is usually interested in computing only the reacting-scalar means and (sometimes) the covariances. For binary mixing in the equilibrium-chemistry limit, these quantities are computed from (5.154) and (5.155), which contain the mixture-fraction PDF. However, since the presumed PDF is uniquely determined from the mixture-fraction mean and variance, (5.154) and (5.155) define mappings (or functions) from (I>- space ... 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

Figure 5.20. The flamelet model requires the existence of unmixed regions in the flow. This will occur only when the mixture-fraction PDF is non-zero at = 0 and = 1. Normally, this condition is only satisfied near inlet zones where micromixing is poor. Beyond these zones, the flamelets begin to interact through the boundary conditions, and the assumptions on which the flamelet model is based no longer apply.

Given a closure for Q(C x, t) and the mixture-fraction PDF, the Reynolds-averaged chemical source term is readily computed ... 

This observation suggests that a moment-closure approach based on the conditional scalar moments may be more successful than one based on unconditional moments. Because adequate models are available for the mixture-fraction PDF, conditional-moment closures focus on the development of methods for finding a general expression for Q( x, t). 

On the other hand, on the bounding hypersurfaces the normal diffusive flux must be null. However, this condition will result naturally from the fact that the conditional joint scalar dissipation rate must be zero-flux in the normal direction on the bounding hypersurfaces in order to satisfy the transport equation for the mixture-fraction PDF.122... 

For a non-premixed homogeneous flow, the initial conditions for (5.299) will usually be trivial Q(C 0 = 0. Given the chemical kinetics and the conditional scalar dissipation rate, (5.299) can thus be solved to find ((pip 0- The unconditional means (y>rp) are then found by averaging with respect to the mixture-fraction PDF. All applications reported to date have dealt with the simplest case where the mixture-fraction vector has only one component. For this case, (5.299) reduces to a simple boundary-value problem that can be easily solved using standard numerical routines. However, as discussed next, even for this simple case care must be taken in choosing the conditional scalar dissipation rate. 

As shown in Chapter 6, the mixture-fraction PDF in a homogeneous flow (f t) obeys a simple transport equation ... 

This boundary condition does not ensure that the unconditional means will be conserved if the chemical source term is set to zero (or if the flow is non-reacting with non-zero initial conditions Q( 0) 0). Indeed, as shown in the next section, the mean values will only be conserved if the conditional scalar dissipation rate is chosen to be exactly consistent with the mixture-fraction PDF. An alternative boundary condition can be formulated by requiring that the first term on the right-hand side of (5.299) (i.e., the diffusive term) has zero expected value with respect to the mixture-fraction PDF. However, it is not clear how this global condition can be easily implemented in the solution procedure for (5.299). 

More specifically, the condition that the probability flux at the boundaries is zero and the condition that the mean mixture-fraction vector is constant in a homogeneous flow lead to natural boundary conditions (Gardiner 1990) for the mixture-fraction PDF governing equation. 

Thus, Z(f, t) is completely determined by / (f, t). As a consequence, as was first pointed out by Tsai and Fox (1995a), the conditional scalar dissipation rate cannot be chosen independently of the mixture-fraction PDF. 

If (5.303) is disregarded and the functional form for the conditional scalar dissipation rate is chosen based on other considerations, an error in the unconditional scalar means will result. Defining the product of the conditional scalar means and the mixture-fraction PDF by... 

In other words, either die mixture-fraction PDF or die conditional reaction-progress vector (but not necessarily both) must be zero on the boundaries of mixture-fraction space. 

The conditional velocity also appears in the inhomogeneous transport equation for x. / ), and is usually closed by a simple gradient-diffusion model. Given the mixture-fraction PDF, (5.316) can be closed in this manner by first decomposing the velocity into its mean and fluctuating components ... 

While inconsistent with the closure for the mixture-fraction PDF, (5.319) does yield the usual gradient-diffusion model for the scalar flux, i.e., for ( ,f). However, it will not predict the correct behavior in certain limiting cases, e.g., when the mixture-fraction mean is constant, but the mixture-fraction variance depends on x.128... 

A two-term power series expression can be derived to handle this case, but it will again fail in cases where the skewness depends on x, but the mean and variance are constant. However, note that the beta PDF can be successfully handled with the two-term form since all higher-order moments depend on the mean and variance. By accounting for the entire shape of the mixture-fraction PDF, (5.318) will be applicable to all forms of the mixture-fraction PDF. 

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