Scalar mean conditional

The scalar mean conditioned on the mixture-fraction vector can be denoted by... 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

Figure 5.21. Scatter plot of concentration in a turbulent reacting flow conditioned on the value of the mixture fraction. Although large fluctuations in the unconditional concentration are present, the conditional fluctuations are considerably smaller. In the limit where the conditional fluctuations are negligible, the chemical source term can be closed using the conditional scalar means. 

If the conditional fluctuations p p are neglected, the homogeneous conditional scalar mean Q(C t) = ( jpIO is governed by (Klimenko 1990 Bilger 1993) (summation is implied with respect to j and k)... 

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... 

Note that (6.190) contains a number of conditional expected values that must be evaluated from the particle fields. The Lagrangian VCIEM model follows from (6.86), and has the same form as the LIEM model, but with the velocity, location-conditioned scalar mean (0 U, X )(U. X. t) in place of location-conditioned scalar mean ( X )(X. t) in the final term on the right-hand side of (6.190). 

Note that A depends only on the conditional scalar means ([Pg.396]

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

Another conditional expectation that frequently occurs in closures for the chemical source term is the conditional mean of the composition variables given the mixturefraction. The latter, defined in Chapter 5, is an inert scalar formed by taking a linear combination of the components of 0 ... 

For perfectly aligned mean scalar gradients, cost 4, ) = 1. Note also that in order for (3.184) to hold, 0 < CaaCpp < C, where the equality holds when the Schmidt numbers are equal. This condition has important ramifications when developing models for... 

Figure 4.14. Predictions of the multi-variate SR model for Re, = 90 and Sc = (1, 1/8) with collinear mean scalar gradients and no backscatter (cb = 0). For these initial conditions, the scalars are uncorrelated pap(0) = gap(0) = 0. The correlation coefficient for the dissipation range, pD, is included for comparison with pap. 

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. 

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). 

The composition PDF thus evolves by convective transport in real space due to the mean velocity (macromixing), by convective transport in real space due to the scalar-conditioned velocity fluctuations (mesomixing), and by transport in composition space due to molecular mixing (micromixing) and chemical reactions. Note that any of the molecular mixing models to be discussed in Section 6.6 can be used to close the micromixing term. The chemical source term is closed thus, only the mesomixing term requires a new model. 

Note, however, that in the presence of a mean scalar gradient the local isotropy condition is known to be incorrect (see Warhaft (2000) for a review of this topic). Although most molecular mixing models do not account for it, the third constraint can be modified to... 

Local mixing is best defined in terms of stochastic models. However, this condition is meant to mle out models based on jump processes where the scalar variables jump large distances in composition space for arbitrarily small df. It also rales out interactions between points in composition space and global statistics such as the mean. 

In order to simulate (6.194) and (6.195) numerically, it will be necessary to estimate the location-conditioned mean scalar field < />. Y )(.v. t) from the notional particles X(ni(j), (p t) for n e 1,..., Nv. In order to distinguish between the estimate and the true value, we will denote the former by

notional particles used in the simulation. Likewise, the subscript M is a reminder that the estimate will depend on the number of grid cells (M) used to resolve the mean fields across the computational domain. 

In the first application of (B.40) to an inhomogeneous bi-variate inert-scalar-mixing case (i.e., the so-called three-stream mixing problem (Juneja and Pope 1996)), it was found that, although the lower-order mixed moments are exactly reproduced, the conditional means (fa) become unrealizable (Marchisio and Fox 2003). Indeed, for every possible choice of the lower-order moments, the sum of the conditional mixture-fraction... 

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