One-point PDF

In words, the right-hand side is the probability that the random variable U (x, t) falls between the sample space values V) and V) + dV) for different realizations of the turbulent flow.5 In a homogeneous flow, this probability is independent of x, and thus we can write the one-point PDF as only a function of the sample space variable and time /(Vi i ). 

Conditional moments of this type cannot be evaluated using the one-point PDF of the mixture fraction alone (O Brien and Jiang 1991). In order to understand better the underlying closure problem, it is sometimes helpful to introduce a new random field, i.e.,15... 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

As mentioned above, the one-point PDF description does not provide the length-scale information needed to predict the decay rate of the scalar variance. For this purpose, a... 

The one-point PDF contains no length-scale information so that scalar dissipation must be modeled. In particular, the mixing time scale must be related to turbulence time scales through a model for 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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