Joint scalar dissipation rate derivation

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. 

The first factor occurs even in homogeneous flows with two inert scalars, and is discussed in Section 3.4. The second factor is present in nearly all turbulent reacting flows with moderately fast chemistry. As discussed in Chapter 4, modeling the joint scalar dissipation rate is challenging due to the need to include all important physical processes. One starting point is its transport equation, which we derive below. 

The transport equation for the joint scalar dissipation rate can be derived following the same steps used to derive (3.110). We begin by defining the fluctuating scalar gradients as... 

A transported PDF extension of the Hamelet model can be derived in a similar manner using the Lagrangian spectral relaxation model (Fox 1999) for the joint scalar dissipation rate. 

In the absence of mean scalar gradients, the scalar covariances and joint dissipation rates will decay towards zero. For this case, it is convenient to work with the governing equations for g p and p p directly. These expressions can be derived from (3.179) and (3.180) ... 

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