FP model

The reasoning behind each of these properties will be illustrated in the next section. We will then look at three simple molecular mixing models (namely, the CD, the IEM, and the FP models) and discuss why each is not completely satisfactory. For convenience, the list of constraints and desirable properties is summarized in Table 6.1. 

Most molecular mixing models concentrate on step (1). However, for chemical-reactor applications, step (2) can be very important since the integral length scales of the scalar and velocity fields are often unequal (L / Lu) due to the feed-stream configuration. In the FP model (discussed below), step (1) is handled by the shape matrix H, while step (2) requires an appropriate model for e. 

However, if the correlation matrix p is rank-deficient, but the scalar dissipation matrix is full rank, the IEM model cannot predict the increase in rank of p due to molecular diffusion. In other words, the last term on the right-hand side of (6.105), p. 278, due to the diffusion term in the FP model will not be present in the IEM model. The GIEM model violates the strong independence condition proposed by Pope (1983). However, since in binary mixing the scalar fields are correlated with the mixture fraction, it does satisfy the weak independence condition. The expected value on the left-hand side is with respect to the joint PDF (c, f x, t). 

In order to go beyond the simple description of mixing contained in the IEM model, it is possible to formulate a Fokker-Planck equation for scalar mixing that includes the effects of differential diffusion (Fox 1999).83 Originally, the FP model was developed as an extension of the IEM model for a single scalar (Fox 1992). At high Reynolds numbers,84 the conditional scalar Laplacian can be related to the conditional scalar dissipation rate by (Pope 2000)... 

This expression does not determine the mixing model uniquely. However, by specifying that the diffusion matrix in the resulting FP equation must equal the conditional joint scalar dissipation rate,88 the FP model for the molecular mixing term in the form of (6.48)... 

This model is consistent with (6.67), and can be seen as a multi-variate version of the IEM model. The role of the second term (eC 1) is simply to compensate for the additional diffusion term in (6.91). Note that, like with the flamelet model and the conditional-moment closure discussed in Chapter 5, in the FP model the conditional joint scalar dissipation rates ( ap ip) must be provided by the user. Since these functions have many independent variables, and can be time-dependent due to the effects of transport and chemistry, specifying appropriate functional forms for general applications will be non-trivial. However, in specific cases where the scalar fields are perfectly correlated, appropriate functional forms can be readily established. We will return to this question with specific examples below. 

Unlike the velocity field, scalar fields are often perfectly correlated so that the correlation matrix p will be rank-deficient.92 When this occurs, the coefficient matrix M will not be properly defined by (6.93), and so the FP model must be modified to handle perfectly correlated scalars. As shown in Section 5.9 for the multi-variate Gaussian presumed PDF,... 

In order to illustrate the properties of the FP model, it is easier to rewrite it in terms of an equivalent stochastic differential equation (SDE) (Arnold 1974 Risken 1984 Gardiner... 

In a numerical implementation of the FP model, Sp is found by replacing eigenvalues which are smaller than some minimum value with one, and there is no need to put the eigenvalues/eigenvectors in descending order. 

The strong independence condition proposed by Pope (1983) will not be satisfied by the FP model unless M is diagonal. This will occur only when all scalar fields are uncorrelated. In this limit, p = I, and e is diagonal. The FP model thus satisfies only the weak independence condition, and scalar-field correlation is essentially determined by the model for e. 

Before looking at its other properties, we should note that by construction the FP model satisfies constraints (I) and (II) (refer again to Table 6.1), and can be made to satisfy constraint (III) by introducing velocity conditioning in all expected values appearing in... 

Property (ii) is also controlled by the behavior of Sg(0)Cg(0). In general, the diffusion matrix should have the property that it does not allow movement in the direction normal to the surface of the allowable region.100 Defining the surface unit normal vector by n(0 ), property (ii) will be satisfied if Sg(0 )Cg(0 )n(0 ) = 0, where 0 lies on the surface of the allowable region. This condition implies that (e 10 )n(0 ) = 0, which Girimaji (1992) has shown to be true for the single-scalar case. Thus, the FP model satisfies property (ii), but the user must provide the unknown conditional joint scalar dissipation rates that satisfy (e 0 )n(0+) = 0. 

Property (iii) applies in the absence of differential-diffusion effects. In this limit, the FP model becomes... 

Like the IEM model, the FP model weakly satisfies property (iv). Likewise, property (v) can be built into the model for the joint scalar dissipation rates (Fox 1999), and the Sc dependence in property (vi) is included explicitly in the FP model. Thus, of the three molecular mixing models discussed so far, the FP model exhibits the greatest number of desirable properties provided suitable functional forms can be found for (e 0). 

The remaining challenge is then to formulate and solve transport equations for the mapping functions g(z x, 0 (Gao and O Brien 1991 Pope 1991b). Note that if g(z x, 0 is known, then the FP model can be used to describe Z(0, and 0(0 will follow from (6.121). Since the PDF of Z is stationary and homogeneous, the FP model needed to describe it will be particularly simple. With the mapping closure, the difficulties associated with the chemical source terms are thus shifted to the model for g(z x, 0. 

Truncating the power series at second order implies that only the means and covariances will be needed to specify the coefficients. The beta PDF has this property, and thus we can speculate that the stationary PDF predicted by the FP model with these coefficients should be the same as (5.147). 

Before leaving the FP model, it is of interest to consider particular limiting cases wherein the form of (e 0) is relatively simple. For example, in many non-premixed flows without differential diffusion, the composition vector is related to the mixture fraction by a linear transformation 107... 

Note that the vector functions go and gi will normally be time-dependent, but can be found from the conditional moments (01 %). In the transported PDF context, the latter can be computed directly from the joint composition PDF so that g0 and gi will be well defined functions.110 The FP model in this limit is thus equivalent to a transported PDF extension of the conditional-moment closure (CMC) discussed in Section 5.8.111 The FP model (including the chemical source term S(0, f)) becomes... 

Because the conditional scalar Laplacian is approximated in the FP model by a non-linear diffusion process (6.91), (6.145) will not agree exactly with CMC. Nevertheless, since transported PDF methods can be easily extended to inhomogeneous flows,113 which are problematic for the CMC, the FP model offers distinct advantages. 

When applying (6.145) and (6.146), the conditional scalar dissipation rate (ej f) must be supplied. However, unlike with the CMC, where both (e f) and a consistent mixture-fraction PDF /f(f) must be provided by the user,114 the FP model predicts the mixture-fraction PDF. Indeed, the stationary115 mixture-fraction PDF predicted by (6.145) is (Gardiner 1990)... 

For inhomogeneous flows, turbulent transport will bring fluid particles with different histories to a given point in the flow. Thus, it cannot be expected that (6.143) will be exact in such flows. Nonetheless, since the conditional moments will be well defined, the FP model may still provide a useful approximation for molecular mixing. 

Using this expression and a model for in (6.145) and (6.146) yields a complete description of reactive scalar mixing without the consistency problems associated with CMC. Moreover, since it is not necessary to supply boundary conditions for the conditional moments,116 the FP model can be applied to partially mixed regions of the flow117 where the CMC boundary conditions cannot be predicted a priori. 

Forthe FP model, the shape information is contained in the shape matrix H(< ), and rate information is contained in die mean joint scalar dissipation rate matrix . 

The LSR model must be applied in conjunction with a consistent Lagrangian mixing model for For example, if the Lagrangian FP model is used, one consistent model has the form... 

Recall that the production term 7 in (6.285) results from the scalar flux, which is not included in (6.287).) As with the FP model discussed in Section 6.6, the doubly conditioned scalar dissipation rate must be supplied by the user. For example, the conditional scalar PDF / (i// t) generated by... 

For higher-order reactions, a model must be provided to close the covariance source terms. One possible approach to develop such a model is to extend the FP model to account for scalar fluctuations in each wavenumber band (instead of only accounting for fluctuations in In any case, correctly accounting for the spectral distribution of the scalar covariance chemical source term is a key requirement for extending the LSR model to reacting scalars. 

In order to close (Jwe can recognize that because J(0) depends only on the 0, it is possible to replace e by (e The closure problem then reduces to finding an expression for the doubly conditioned joint scalar dissipation rate matrix. For example, if the FP model is used to describe scalar mixing, then a model of the form... 

The term A2Pr is a direct result of employing the IEM model. If a different mixing model were used, then additional terms would result. For example, with the FP model the right-hand side would have the form /3 = A3PC + A2Pr + AsPra, where rd results from the diffusion term in the Fokker-Planck equation. 

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