Particle-field estimation

Lagrangian mixing models Particle-field estimation... 

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. 

We will look next at the specific algorithms needed to advance the PDF code. In particular, we describe the MC simulation needed to advance the particle position, the application of boundary conditions, and particle-field estimation. We then conclude our discussion of Lagrangian composition PDF codes by considering other factors that can be used to obtain simulation results more efficiently. 

At the end of the chemical-reaction step, all particle properties (w n>, X(n), fl(n>) have been advanced in time to t + At. Particle-field estimates of desired outputs can now be constructed, and the MC simulation is ready to perform the next time step. For a constant-density flow, the particle-field estimates are not used in the FV code. Thus, for stationary flow, the particle properties can be advanced without returning to the FV code. For unsteady or variable-density flow, the FV code will be called first to advance the turbulence fields before calling the PDF code (see Fig. 7.3). 

In general, the estimated mean composition field will depend on the form of co(x,t) (see, for example, Jenny etal. (2001)). Thus, care must be taken when defining X in order to ensure that (7.36) conserves the mean compositions. We will look at this point in detail when discussing particle-field estimation below. 

Since the particles are randomly located in grid cells, interpolation and particle-field-estimation algorithms are required. Special care is needed to ensure local mass conservation (i.e., continuity) and to eliminate bias. 

Each of these steps introduces numerical errors that must be carefully controlled (Jenny et al. 2001). In particular, taking spatial gradients in step (2) using noisy estimates from step (1) can lead to numerical instability. We will look at this question more closely when considering particle-field estimation below. 

See Jenny el al. (2001) for a discussion of particle-field estimation for variable-density flows. 

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