Notional particles

The numerical methods employed to solve the transported PDF transport equation are very different from standard CFD codes. In essence, the joint PDF is represented by a large collection of notional particles. The idea is similar to the presumed multi-scalar PDF method discussed in the previous section. The principal difference is that the notional particles move in real and composition space by well defined stochastic models. Some of the salient features of transported PDF codes are listed below. 

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. 

A Lagrangian notional particle follows a trajectory in velocity-composition-physical space (i.e., U O), X (r))122 which originates at a random location Y in the physical... 

We will also discuss Lagrangian PDF models for the composition PDF. In this case, the notional particles follow... 

A more precise definition would include conditioning on the random initial velocity and compositions /li, , x Uo,. o.Y Vb XIY), V o, y 0- However, only the conditioning on initial location is needed in order to relate the Lagrangian and Eulerian PDFs. Nevertheless, the initial conditions (Uo, o) for a notional particle must have the same one-point statistics as the random variables U(Y, to) and (V. to). 

B B. By correctly choosing the coefficient matrices (a u, a 7, By, and B ), (6.159) can be made to correspond with the Eulerian velocity, composition PDF transport equation (6.19). However, it is important to note that /L / Thus it remains to determine how the Lagrangian notional-particle PDF /,( is related to the Eulerian velocity, composition PDF fn,0. This can be done by considering Lagrangian fluid particles. 

For a Lagrangian PDF model, (6.162) should hold when /u, , x y is substituted for. /1j1,1,x11y- Thus, we will now look at what conditions are necessary to ensure that (6.162) holds for the notional-particle PDF. 

As shown above in (6.162), the Lagrangian fluid-particle PDF can be related to the Eulerian velocity, composition PDF by integrating over all initial conditions. As shown below in (6.168), for the Lagrangian notional-particle PDF, the same transformation introduces a weighting factor which involves the PDF of the initial positions y) and the PDF of the current position /x.(x t). If we let V denote a closed volume containing a fixed mass of fluid, then, by definition, x, y e V. The first condition needed to reproduce the Eulerian PDF is that the initial locations be uniform ... 

It is important to recognize that (6.164) is a direct result of choosing the uniformly distributed initial locations (6.163). In contrast, if one chooses to start all notional particles at the origin /V(y) = <5(y), then /x (x t) will be non-uniform, and the Lagrangian notional-particle PDF will not correspond to the Lagrangian fluid-particle PDF. 

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... 

In words, the Eulerian PDF generated by the notional particles is equal to the Lagrangian PDF for velocity and composition at a given spatial location ... 

If the initial particle locations are uniformly distributed and (6.160) holds, then (6.167) relates the Eulerian notional-particle PDF to the Eulerian PDF (for fixed x and t with... 

Furthermore, since (6.159) does not depend on y, if the notional particles are uniformly distributed the Fokker-Planck equation for f is... 

Thus, in summary, the two necessary conditions for correspondence between the notional-particle system and the fluid-particle system in constant-density flows are... 

Thus, correspondence between the notional-particle system and the Eulerian PDF of the flow requires agreement at the moment level. In particular, it requires that (U(x, /)) = (U (r) X (0 = x) and ((x, t)) = (0 (r) X (O = x). It remains then to formulate stochastic differential equations for the notional-particle system which yield the desired correspondence. 

Using (6.4) and (6.6), the notional-particle trajectories can be expressed in terms of the conditional fluxes 131 dX ... 

In general, hie conditional fluxes are deterministic functions of V, x, and t. On hie right-hand side of these equations, the conditional fluxes are evaluated at hie current location of the notional particle in velocity-composition-physical space V = U (0, tp = ), x = X ( ). 

Note that the turbulent diffusivity Tt(x, t) must be provided by a turbulence model, and for inhomogeneous flows its spatial gradient appears in the drift term in (6.177). If this term is neglected, the notional-particle location PDF, fx>, will not remain uniform when VTt / 0, in which case the Eulerian PDFs will not agree, i.e., i=- f0. 

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... 

The superscript used in the coefficient matrices in (6.192) is a reminder that the statistics must be evaluated at the notional-particle location. For example, e = e(X r), and the scalar standard-deviation matrix and scalar correlation matrix p are computed from the location-conditioned scalar second moments X )(X, t). 

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 the SDEs, we will introduce a large ensemble of notional particles that move through the simulation domain according to the Lagrangian PDF models. As an example, we will consider a single inert-scalar field in a one-dimensional domain. The position and composition of the th notional particle can be denoted by X n t) and 4i(n)(f), respectively. The SDEs for the Lagrangian composition PDF (with closures) become... 

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

grid cells (M) used to resolve the mean fields across the computational domain. 

In general, will scale as A p and thus can only be eliminated by increasing the number of notional particles. The quantity ( < > X NptM) can be estimated by running multiple independent simulations with fixed Np and M (Xu and Pope 1999). 

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. 

Note that, by construction, all notional particles are identically distributed. Thus, in the absence of deterministic errors caused by using

true mean field, the Lagrangian PDF (/ w.xw) found from (6.194) and (6.195) would be equal... 

In order to use the notional particles to estimate f x, we need a method to identify a finite sample of notional particles in the neighborhood of x on which to base our estimate. In transported PDF codes, this can be done by introducing a kernel function hw(s) centered at 5 = 0 with bandwidth W. For example, a so-called constant kernel function (Wand and lones 1995) can be employed ... 

Using the kernel function, we can define the number of notional particles at point x by... 

With A/(x), the right-hand side will just be the sum over all notional particles in cell / divided by Wp/. 

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