Joint scalar dissipation rate fluctuating

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... 

We next define the fluctuating joint scalar dissipation rate rap by... 

The model proposed by Fox (1999) also accounts for fluctuations in the joint scalar dissipation rate. Here we will look only at the simpler case, where the is deterministic. 

The scalar dissipation rate acts as an external parameter that is imposed on the flamelet structure by the mixture fraction field [15]. It describes the influence of the turbulent flow field on the laminar flame stracture. Both the mixture fraction and the scalar dissipation rate fluctuate on turbulent flows, and their statistical distribution needs to be considered. If the joint pdf P Z, Xst) (where Xst is x at the stoichiometric condition) is known, the Favre mean of 7 can be obtained from... 

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. 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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