Scalar dissipation rate

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 ... 

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... 

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-... 

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) ... 

Equation (3.82) illustrates the importance of the scalar spectral energy transfer rate in determining the scalar dissipation rate in high-Reynolds-number turbulent flows. Indeed, near spectral equilibrium, 7 (/cd, 0 (like Tu(kDi, 0) will vary on time scales of the order of the eddy turnover time re, while the characteristic time scale of (3.82) is xn <[Pg.99]

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

Thus, (3.105) has three unclosed terms the scalar flux Uj), the scalar variance flux (Uj(p/2), and the scalar dissipation rate e, defined by... 

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 transport equation for the scalar dissipation rate of an inert scalar can be derived starting from (3.90). We begin by defining the fluctuating scalar gradient as... 

Next we define the fluctuating scalar dissipation rate by... 

Note that in the turbulent mixing literature, the scalar dissipation rate is often defined without the factor of two in (3.112). Likewise, in the combustion literature, the symbol X is used in place of in (3.112). In this book, we will consistently use to denote the fluctuating scalar dissipation rate, and = (e ) to denote the scalar dissipation rate. 

Multiplying (3.111) by 2Y and Reynolds averaging yields the final form for the scalar-dissipation-rate transport equation ... 

Combining (3.122) and (3.127), the scalar dissipation rate for homogeneous turbulent mixing can be expressed as31... 

For large Reynolds numbers, the right-hand side of this expression will be large, thereby forcing the scalar dissipation rate to attain a stationary solution quickly. Thus, for a fully developed scalar spectrum, the scalar mixing rate is related to the turbulent frequency by... 

Using (3.130), the transport equation for the scalar dissipation rate in high-Reynolds-number homogeneous turbulence becomes... 

Thus, like the turbulence dissipation rate, the scalar dissipation rate of an inert scalar is primarily determined by the rate at which spectral energy enters the scalar dissipation range. Most engineering models for the scalar dissipation rate attempt to describe (kd, t) in terms of one-point turbulence statistics. We look at some of these models in Chapter 4. 

The dissipation term35 in (3.136) is written as the product of the joint scalar dissipation rate eap defined by... 

Unlike the turbulence dissipation rate tensor, which is isotropic at high Reynolds number, the joint scalar dissipation rate tensor is usually highly anisotropic. Indeed, when r< = T, it is often the case for inert scalars that eap = eaa = , so that the joint scalar dissipation rate tensor is singular. 

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