Conditional scalar dissipation rate

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  

121 This boundary condition does not ensure that the unconditional means will be conserved if the chemical source 

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

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. 

---

In other closures for the chemical source term, a model for the conditional scalar dissipation rate (e

scalar Laplacian, the conditional scalar... 

Figure 1.13. The conditional scalar dissipation rate for the scalar PDF in Fig. 1.11.

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). 

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... 

Note that if the conditional scalar dissipation rate is chosen correctly (i.e., Z = Z), then the first term on the right-hand side of this expression is null. However, if Z is inconsistent with /f, then the scalar means will be erroneous due to the term... 

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)... 

Applying the same procedure to higher-dimensional mixture-fraction vectors yields expressions of the same form as (6.130). Note also that for any set of bounded scalars that can be linearly transformed to a mixture-fraction vector, (6.115) can be used to find the corresponding joint conditional scalar dissipation rate matrix starting from (e% C). 

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)... 

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... 

---