Variance mixture-fraction

Thus, the reactor will be perfectly mixed if and only if = at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by ( ). Thus, the first state variable needed to describe macromixing in this system is ( ). If the system is perfectly macromixed, ( ) = < at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance (maximum value of the variance at any point in the reactor is ( )(1 — ( )), and varies from zero in the feed streams to a maximum of 1/4 when ( ) = 1/2. 

In the limit where the mixture-fraction-variance dissipation rate is null, the mixture-fraction variance is related to the mean mixture fraction by... 

In this limit, the mixing time scale (r ) is infinite and thus (5.61) predicts correctly that (SA) is null. In the other limit where is null, the mixture-fraction variance will also be null, so that (5.61) again predicts the correct limiting value for the mean chemical source term. Between these two limits, the magnitude of (SA) will be decreased due to the finite... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

Figure 5.16. The four-environment model divides a homogeneous flow into four environments, each with its own local concentration vector. The joint concentration PDF is thus represented by four delta functions. The area under each delta function, as well as its location in concentration space, depends on the rates of exchange between environments. Since the same model can be employed for the mixture fraction, the exchange rates can be partially determined by forcing the four-environment model to predict the correct decay rate for the mixture-fraction variance. The extension to inhomogeneous flows is discussed in Section 5.10.

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... 

Comparing (5.377) with (3.105) on p. 85 in the high-Reynolds-number limit (and with e = 0), it can be seen that (5.378) is a spurious dissipation term.149 This model artifact results from the presumed form of the joint composition PDF. Indeed, in a transported PDF description of inhomogeneous scalar mixing, the scalar PDF relaxes to a continuous (Gaussian) form. Although this relaxation process cannot be represented exactly by a finite number of delta functions, Gs and M1 1 can be chosen to eliminate the spurious dissipation term in the mixture-fraction-variance transport equation.150... 

The agreement between higher-order moments will not be assured. However, this is usually not as important as agreement with the mixture-fraction variance, which controls the rate of micromixing relative to chemical reactions. A complete treatment of this problem is given in Appendix B. 

The mixture-fraction-variance transport equation can be found starting from (5.383) and (5.384) 151... 

Note that the effect of the spurious dissipation term can be non-trivial. For example, consider the case where at t — 0 the system is initialized with p = 1, but with ( )i varying as a function of x. The micromixing term in (5.387) will initially be null, but ys will be non-zero. Thus, p2 will be formed in order to generate the correct distribution for the mixture-fraction variance. By construction, the composition vector in environment 2 will be constant and equal to )2. 

For this reason, mixing in the conditional PDF model is orthogonal to mixing in the unconditional model. Thus, y in the conditional model need not be the same as in the unconditional model, where its value controls tiie mixture-fraction-variance decay rate. 

The model for the residual mixture-fraction variance found from (5.421) and (5.422) has the form... 

The mixing parameter Q must be chosen to yield the correct mixture-fraction-variance dissipation rate. However, inertial-range scaling arguments suggest that its value should be near unity.165... 

The DQMOM results for the IEM model can be compared with the multi-environment presumed PDF models in Section 5.10. In particular, (5.374) on p. 226 can be compared with (B.44), and (5.375) can be compared with (B.45). First, we can note that for the IEM model G = 0 and Gs = a. Likewise, y AT"1 + pnSa 4>)n) = 7 .,z and = Ban. Of the four models introduced in Tables 5.1-5.5, only the symmetric two-environment model in Table 5.2 has G = 0 and yM(an) + pnSa = 7Zan. However, because the spurious dissipation terms only ensure that the mixture-fraction variance is correctly predicted, the symmetric two-environment model does not have Gs = a and Mg 1 = Ban. Thus, the covariance matrix is not predicted correctly, as it would be if (B.43) were used. We can thus conclude that the multi-environment presumed PDF models are incomplete in the sense that they do not control as many of the moments as possible for a given choice of -/Ve. 

Thus, PDF and time-averaged values of individual species can be calculated if the mean values of mixture fraction (f) and mixture fraction variance (f ) are known. 

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