Scalar dissipation rate equilibrium model

The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

As discussed in Chapter 4, the modeling of the scalar dissipation rate in (3.105) is challenging due to the need to describe both equilibrium and non-equilibrium spectral... 

The second unclosed term is the scalar dissipation rate e. The most widely used closure for this term is the equilibrium model (Spalding 1971 Beguier et al. 1978) ... 

In the definition of b, Ro is the equilibrium mechanical-to-scalar time-scale ratio found with Sc = 1 and = 0.34 The parameters Cd, Cb, and Cd appear in the SR model for the scalar dissipation rate discussed below. Note that, by definition, xi + Yi + K3 + Kd = 1. ... 

Application of the SLF model thus reduces to predicting the joint PDF of the mixture fraction and the scalar dissipation rate. As noted above, in combusting flows flame extinction will depend on the value of x Thus, unlike the equilibrium-chemistry method (Section 5.4), the SLF model can account for flame extinction due to local fluctuations in the scalar dissipation rate. 

Cs = Cb - Co, Cb = 1, and Cd = 3 (Fox 1995).36 Note that at spectral equilibrium, Vp = p, % = To = p( I - i/i)), and (with Sc = 1) R = Rq. The right-hand side of (4.117) then yields (4.114). Also, it is important to recall that unlike (4.94), which models the flux of scalar energy into the dissipation range, (4.117) is a true small-scale model for p. For this reason, integral-scale terms involving the mean scalar gradients and the mean shear rate do not appear in (4.117). Instead, these effects must be accounted for in the model for the spectral transfer rates. 

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