Scalar dissipation

Qualitative comparison of the inclined structure of thin layers of high scalar dissipation in a piloted CH4/air jet flame as revealed by (a) mixture fraction imaging, (b) LES with a steady flamelet library (a and b are adapted from Kempf, A. Flemming, F., and Janicka, ]., Proc. Combust. Inst, 30, 557, 2005. With permission.), and (c) LES with unsteady flamelet modeling. (Adapted from Pitsch, H. and Steiner, H., Proc. Combust. Inst., 28, 41, 2000. With permission.)... 

Kempf, A., Flemming, F, and Janicka, J., Investigation of lengthscales, scalar dissipation, and flame orientation in a piloted diffusion flame by LES, Proc. Combust. Inst., 30, 557, 2005. 

Volume rendering of scalar dissipation rate in a DNS of a temporally evolving CO/Hj jet flame. Re = 9200 [16]. The highest values of scalar dissipation rate (shown in red) exceed 30,000 S . ... 

The degree of local mixing in a RANS simulation is measured by the scalar variance (complete mixing (i.e., (j> — (j>) is uniform at the SGS) up to (4>max — (4>))((4>) — 4>min) where () is the mean concentration and max and r/>min are the maximum and minimum values, respectively. The rate of local mixing is controlled by the scalar dissipation rate (Fox, 2003). The scalar time scale analogous to the turbulence integral time scale is (Fox, 2003) as follows ... 

Note that the scalar-dissipation constant computed from Eq. (21) depends only on ReL and Sc. 

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). 

In Section 3.2, we show that under the same conditions the right-hand side of (1.24) is equal to the negative scalar dissipation rate ((3.45), p. 70). Thus, the micromixing time is related to the scalar dissipation rate e and the scalar variance by... 

Choosing the micromixing time in a CRE micromixing model is therefore equivalent to choosing the scalar dissipation rate in a CFD model for scalar mixing. 

As seen above, the mean chemical source term is intimately related to the PDF of the concentration fluctuations. In non-premixed flows, the rate of decay of the concentration fluctuations is controlled by the scalar dissipation rate. Thus, a critical part of any model for chemical reacting flows is a description of how molecular diffusion works to damp out... 

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.

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

In developing closures for the chemical source term and the PDF transport equation, we will also come across conditional moments of the derivatives of a field conditioned on the value of the field. For example, in conditional-moment closures, we must provide a functional form for the scalar dissipation rate conditioned on the mixture fraction, i.e.,... 

Likewise, the scalar dissipation rate is related to the scalar energy spectrum by... 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

Figure 3.11. The scalar-dissipation constant cd found with the Kraichnan cut-off function as a function of Schmidt number at various Reynolds numbers Rx = 50, 100, 200, 400, and 800. The arrow indicates the direction of increasing Reynolds number.

However, DNS data for Schmidt numbers near unity suggest that (3.70) provides the best model for the scalar-dissipation range (Yeung et al. 2002). 

The scalar-dissipation constant cd appearing in (3.69) and (3.70) is fixed by forcing the integral of the scalar-dissipation spectrum to satisfy (3.54) 20... 

Thus, the final term in (3.73) is responsible for scalar dissipation due to molecular diffusion at wavenumbers near /cB-... 

In a fully developed turbulent flow,22 the scalar spectral transfer rate in the inertial-convective sub-range is equal to the scalar dissipation rate, i.e., T k) = for /cei < < Kn. Likewise, when Sc 1, so that a viscous-convective sub-range exists, the scalar trans-... 

For a passive scalar, the turbulent flow will be unaffected by the presence of the scalar. This implies that for wavenumbers above the scalar dissipation range, the characteristic time scale for scalar spectral transport should be equal to that for velocity spectral transport tst defined by (2.67), p. 42. Thus, by equating the scalar and velocity spectral transport time scales, we have23 t)... 

Following the approach used to derive (2.75), p. 43, the scalar spectral transport equation can also be used to generate a spectral model for the scalar dissipation rate for the case 1 < Sc.24 Multiplying (3.73) by 2T/< 2 yields the spectral transport equation for D Ik, t) ... 

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