Transported PDF simulations

We have seen that the joint velocity, composition PDF treats both the velocity and the compositions as random variables. However, as noted in Section 6.1, it is possible to carry out transported PDF simulations using only the composition PDF. By definition, x, t) can be found from /u,< >(V, 0 x, t) using (6.3). The same definition can be used with the transported PDF equation derived in Section 6.2 to find a transport equation for / (0 x, r). [Pg.268]

As with all statistical methods, the mean-field estimate will have statistical error due to the finite sample size (X ), and deterministic errors due to the finite grid size (S ) and feedback of error in the coefficients of the SDEs Ui,p). Since error control is an important consideration in transported PDF simulations, we will now consider a simple example to illustrate the tradeoffs that must be made to minimize statistical error and bias. The example that we will use corresponds to (6.198), where the exact solution141 to the SDEs has the form ... [Pg.321]

ODE solver. Relative to non-stiff ODE solvers, stiff ODE solvers typically use implicit methods, which require the numerical inversion of an Ns x Ns Jacobian matrix, and thus are considerably more expensive. In a transported PDF simulation lasting T time units, the composition variables must be updated /Vsm, = T/At 106 times for each notional particle. Since the number of notional particles will be of the order of A p 106, the total number of times that (6.245) must be solved during a transported PDF simulation can be as high as A p x A sim 1012. Thus, the computational cost associated with treating the chemical source term becomes the critical issue when dealing with detailed chemistry. [Pg.328]

In a transported PDF simulation a large ensemble of notional particles is employed in order to estimate the mean fields accurately 152... [Pg.328]

In transported PDF simulations, this problem can be circumvented by employing... [Pg.328]

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... [Pg.329]

An example of a smart tabulation method is the intrinsic, low-dimensional manifold (ILDM) approach (Maas and Pope 1992). This method attempts to reduce the number of dimensions that must be tabulated by projecting the composition vectors onto the nonlinear manifold defined by the slowest chemical time scales.162 In combusting systems far from extinction, the number of slow chemical time scales is typically very small (i.e, one to three). Thus the resulting non-linear slow manifold ILDM will be low-dimensional (see Fig. 6.7), and can be accurately tabulated. However, because the ILDM is non-linear, it is usually difficult to find and to parameterize for a detailed kinetic scheme (especially if the number of slow dimensions is greater than three ). In addition, the shape, location in composition space, and dimension of the ILDM will depend on the inlet flow conditions (i.e., temperature, pressure, species concentrations, etc.). Since the time and computational effort required to construct an ILDM is relatively large, the ILDM approach has yet to find widespread use in transported PDF simulations outside combustion. [Pg.331]

This formulation can be compared with the multi-environment (ME) presumed PDF method discussed in Section 5.10. The principal difference between the two approaches is the treatment of turbulent convection. In the transported PDF simulation, turbulent convection is simulated by a random process. In the ME—PDF approach, it is handled using standard FV/FD discretization. [Pg.350]

However, in practice, the integral must be evaluated using a stiff ODE solver or chemical lookup tables (see Section 6.9). Because transported PDF simulations are typically used for reacting flows with complex chemistry, the chemical-reaction step will often dominate the overall computational cost. It is thus important to consider carefully the computational efficiency of the chemical-reaction step when implementing a transported PDF simulation. [Pg.365]

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