PDF simulation codes

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. 

Figure 7.6. Coupling between finite-volume and PDF codes in a velocity, composition PDF simulation.

The PDF codes presented in this chapter can be (and have been) extended to include additional random variables. The most obvious extensions are to include the turbulence frequency, the scalar dissipation rate, or velocity acceleration. However, transported PDF methods can also be applied to treat multi-phase flows such as gas-solid turbulent transport. Regardless of the flow under consideration, the numerical issues involved in the accurate treatment of particle convection and coupling with the FV code are essentially identical to those outlined in this chapter. For non-orthogonal grids, the accurate implementation of the particle-convection algorithm is even more critical in determining the success of the PDF simulation. 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

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. 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

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). 

The algorithms discussed earlier for time averaging and local time stepping apply also to velocity, composition PDF codes. A detailed discussion on the effect of simulation parameters on spatial discretization and bias error can be found in Muradoglu et al. (2001). These authors apply a hybrid FV-PDF code for the joint PDF of velocity fluctuations, turbulence frequency, and composition to a piloted-jet flame, and show that the proposed correction algorithms virtually eliminate the bias error in mean quantities. The same code... 

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