Composition PDF

The application of DQMOM to the closed composition PDF transport equation is described in detail by Fox (2003). If the IEM model is used to describe micromixing and a gradient-diffusivity model is used to describe the turbulent fluxes, the CFD model will have the form... 

The final term in Eq. (86) is the correction term b n, which comes from applying DQMOM to the transport equation for the composition PDF (Fox,... 

This expression results from forcing the transport equations for the first N moments of (f>a, denoted by to agree with the composition PDF transport equation (Fox, 2003). For example, with N —2 the linear system in Eq. (89) can be written in matrix form as... 

For fixed initial conditions, the solution to this expression is uniquely defined in terms of the age, i.e., 0batch(oO. The joint composition PDF ftime-dependent RTD distribution 14... 

For the general case of interacting fluid elements, (1.9) and (1.10) no longer hold. Indeed, the correspondence between the RTD function and the composition PDF breaks down because the species concentrations inside each fluid element can no longer be uniquely parameterized in terms of the fluid element s age. Thus, for the general case of complex chemistry in non-ideal reactors, a mixing theory based on the composition PDF will be more powerful than one based on RTD theory. 

Heuristically, the SGS distribution of a scalar field 0(x, t) can be used to estimate the composition PDF by constructing a histogram from all SGS points within a particular CFD grid cell.30 Moreover, because the important statistics needed to describe a scalar field (e.g., its expected value (0) or its variance (e//2)) are nearly constant on sub-grid... 

Because the random velocity field U(x, t) appears in (1.28), p. 16, a passive scalar field in a turbulent flow will be a random field that depends strongly on the velocity field (Warhaft 2000). Thus, turbulent scalar mixing can be described by a one-point joint velocity, composition PDF /u,< (V, i/r,x, t) defined by... 

Nevertheless, in many cases, a simpler description based on the one-point composition PDF f fix, r), defined by... 

The scalar fields appearing in Figs. 3.7 to 3.9 were taken from the same DNS database as the velocities shown in Figs. 2.1 to 2.3. The one-point joint velocity, composition PDF found from any of these examples will be nearly Gaussian, even though the temporal and/or spatial variations are distinctly different in each case.11 Due to the mean scalar... 

Gaussian PDFs are found for homogeneous inert scalar mixing in the presence of a uniform mean scalar gradient. However, for turbulent reacting flows, the composition PDF is usually far from Gaussian due to the non-linear effects of chemical reactions. 

Figure 3.10. Evolution of the inert composition PDF for binary mixing.

Starting from these initial conditions, the composition PDF will evolve in a non-trivial manner due to turbulent mixing and molecular diffusion.13 This process is illustrated in Fig. 3.10, where it can be seen that the shape of the composition PDF at early and intermediate times is far from Gaussian.14 As discussed in Chapter 6, one of the principal challenges in transported PDF methods is to develop mixing models that can successfully describe the change in shape of the composition PDF due to molecular diffusion. 

Turbulent mixing is primarily responsible for fixing the rate at which the composition PDF evolves in time. Molecular diffusion, on the other hand, determines the shape of the composition PDF at different time instants. In fact, binary mixing of an inert scalar is well represented by a beta PDF. 

Table 3.1. The relationship between the chemical-reaction-engineering description of turbulent mixing and the one-point composition PDF description. 

In Section 3.3, we will use (3.16) with the Navier-Stokes equation and the scalar transport equation to derive one-point transport equations for selected scalar statistics. As seen in Chapter 1, for turbulent reacting flows one of the most important statistics is the mean chemical source term, which is defined in terms of the one-point joint composition PDF +(+x, t) by... 

Thus, in Chapter 6, the transport equations for /++ x, t) and the one-point joint velocity, composition PDF /u+V, + x. / ) are derived and discussed in detail. Nevertheless, the computational effort required to solve the PDF transport equations is often considered to be too large for practical applications. Therefore, in Chapter 5, we will look at alternative closures that attempt to replace /++ x, t) in (3.24) by a simplified expression that can be evaluated based on one-point scalar statistics that are easier to compute. 

In cases where this approximation is acceptable, it thus suffices to provide a model for (0 f) in place of a model for the joint composition PDF. Approaches based on this... 

Alternatively, an LES joint velocity, composition PDF can be defined where both (j> andU are random variables Aj 0 U 4 U >4 x, t). In either case, the sample space fields U and0 are assumed to be known. 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

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