Particle-field estimation notional particles

In Section 6.8 we will discuss how particle fields such as (U U X > can be estimated from the notional particles. However, it is important to note that since the particle-pressure field is found by solving (6.179), the estimate of (U U X must be accurate enough to allow second-order derivatives. As noted after (6.61), the problem of dealing with noisy estimates of P(x, t) is one of the key challenges in applying (6.178).134... 

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. 

In order to simulate (6.194) and (6.195) numerically, it will be necessary to estimate the location-conditioned mean scalar field < />. Y )(.v. t) from the notional particles X(ni(j), (p t) for n e 1,..., Nv. In order to distinguish between the estimate and the true value, we will denote the former by

notional particles used in the simulation. Likewise, the subscript M is a reminder that the estimate will depend on the number of grid cells (M) used to resolve the mean fields across the computational domain. 

Ideally, one would like to choose Np and M large enough that e is dominated by statistical error (X ), which can then be reduced through the use of multiple independent simulations. In any case, for fixed Np and M, the relative magnitudes of the errors will depend on the method used to estimate the mean fields from the notional-particle data. We will explore this in detail below after introducing the so-called empirical PDF. 

In a transported PDF simulation a large ensemble of notional particles is employed in order to estimate the mean fields accurately 152... 

The random selection in step (iii) is carried out by generating uniform random numbers U e [0, 1], For example, the index of a random particle selected from a set of N particles will be n = intup(//N) where intuP() rounds the argument up to the nearest integer. Note that for constant-density, statistically stationary flow, the effective flow rates will be constant. In this case, steps (i) and (ii) must be completed only once, and the MC simulation is advanced in time by repeating step (iii) and intra-cell processes. For variable-density flow, the mean density field ((p)) must be estimated from the notional particles and passed back to the FV code. In the FV code, the non-uniform density field is held constant when solving for the mean velocity field.15... 

Although not denoted explicitly, we have seen in Section 6.8 that this estimate will depend on the grid spacing M and the number of particles Nv. In addition to the mean composition, the output data from the PDF code will usually be various composition statistics estimated at grid-cell centers. We will thus need accurate and efficient statistical estimators for determining particle fields given the ensemble of Nv notional particles. 

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