Lagrangian PDF methods

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

The key theoretical concept that makes Lagrangian PDF methods useful is the correspondence between the Eulerian PDF of the flow and the Lagrangian notional-particle PDF. As noted above, in the Lagrangian notional-particle PDF X is a random variable... 

As noted above, in the applications of Lagrangian PDF methods to inhomogeneous flows, evaluation of the particle-pressure field can be problematic. In order to avoid this difficulty, hybrid PDF methods have been developed (Muradoglu, et al. 1999 Jenny et al. 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

Pope, S.B. 1994. Lagrangian PDF methods for turbulent flows. Annual Reviews Fluid Mechanics 26 23-63. 

Prospects for mixing model improvements Lagrangian PDF methods... 

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