Scalar dissipation rate fluctuating

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... 

Pitsch H. Improved pollutant predictions in large-eddy simulations of turbulent non-premixed combustion by considering scalar dissipation rate fluctuations. Proc Combust Inst... 

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... 

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. 

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

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 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. 

Given a stochastic model for the turbulence frequency, it is natural to enquire how fluctuations in co will affect the scalar dissipation rate (Anselmet and Antonia 1985 Antonia and Mi 1993 Anselmet et al. 1994). In order to address this question, Fox (1997) extended the SR model discussed in Section 4.6 to account for turbulence frequency fluctuations. The resulting model is called the Lagrangian spectral relaxation (LSR) model. The LSR model has essentially the same form as the SR model, but with all variables conditioned on the current and past values of the turbulence frequency [ /(. ),. v < r. In order to simplify the notation, this conditioning is denoted by ( , e.g.,... 

The flamelet concept for turbulent combustion applies when the reaction is fast compared to the mixture at the molecular level. In this regime, the chemistry of a flame and the turbulence can be treated separately. The flamelet concept approaches the solution of Burke-Schumann for a high Damkohler number and mechanism of one step. The scalar dissipation rate, which appears in the flamelet equations, relates the effects caused by the diffusion and convection. This rate is large at the smallest scales, but its fluctuations are mainly governed by the large scales, which are solved using Large-Eddy Simulation (LES). 

These equations complete the Lagrangian flamelet model. A transformation of coordinates different from that presented in Eqs. (5.75)-(5.77) results in the Eulerian flamelet model proposed by Pitsch [18]. In the Eulerian system, both velocity vector and scalar dissipation rate are functions of time, space, and the mixture fraction. The difference between these models appears to be the manner in which the fluctuations are taken into account. Because the differences are small, the Lagrangian flamelet model is more employed, because it is easier to implement and represents well the majority applications for diffusion flames. 

For gases, 5c 1, for hquids. Sc 1. This implies that in turbulent flow of liquids, the species concentration field contains smaller scale structures than the velocity field. Similar to the decay time of the turbulent eddies in the velocity field, Tu, (12.2-1), the decay time of the eddies in the species concentration field, the previously introduced micro-mixing time, xy, can be modeled in terms of the correlation of the species mass fraction fluctuations, the so-called scalar (co-) variance, (K F), and its dissipation rate, the so-called scalar dissipation rate, sy. 

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. 

Thus analogously to the inertial scale of turbulence, the statistical properties of the scalar fluctuations in the inertial-convective range, i.e. in a range of scales below the forcing scale where both diffusion and viscosity are negligible, can only depend on the dissipation rate ee, the energy dissipation rate e, and on the length scale. Thus the only dimensionally correct form of the second order scalar structure function of the concentration fluctuations is... 

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenological, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section. 

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