Composition PDF codes

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

In a (constant-density) Lagrangian composition PDF code, the position of a notional particle is governed by... 

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. 

In the Lagrangian composition PDF code, the mixing model requires estimated statistics for the compositions. For example, the LIEM model requires an estimate for the mean composition at the particle location ... 

In a variable-density PDF code, passing back the time-averaged particle fields should have an even greater effect on bias. For example, using a Lagrangian velocity, composition PDF code, Jenny et al. (2001) have shown that the bias error is inversely proportional to the product Vp K. 

In summary, the advantages and disadvantages of Lagrangian composition PDF codes relative to other methods are as follows ... 

Relative to velocity, composition PDF codes, the turbulence and scalar transport models have a limited range of applicability. This can be partially overcome by using an LES description of the turbulence. However, consistent closure at the level of second-order RANS models requires the use of a velocity, composition PDF code. 

While some of these disadvantages can be overcome by devising improved algorithms, the problem of level of description of the RANS turbulence model remains as the principal shortcoming of composition PDF code. One thus has the option of resorting to an LES description of the flow combined with a composition PDF code, or a less-expensive second-order RANS model using a velocity, composition PDF code. 

In a velocity, composition PDF code, each notional particle is described by its position Xw(f), fluctuating velocity uw(f), composition and weight w(n)(t). The treatment... 

A velocity, composition PDF code is consistent when the fields predicted by the FV code and the PDF code are in agreement (Muradoglu et al. 1999 Muradoglu el al. 2001). For the constant-density algorithm described above, two independent conditions must be met to ensure consistency.50... 

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... 

Relative to Lagrangian composition PDF codes that use an LES description of the flow, the turbulence models used in velocity, composition PDF codes have a limited range of applicability. However, the computational cost of the latter for reacting flows with detailed chemistry will be considerably lower. 

As with any numerical algorithm, it is important to test transported PDF codes for accuracy using known solutions. Typical test cases for composition PDF codes include... 

The same test cases can be used for velocity, composition PDF codes, but with the turbulent kinetic energy and turbulence frequency specified in place of the turbulent diffusivity. 

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