LES velocity PDF

The filtering process used in LES results in a loss of information about the SGS velocity field. For homogeneous turbulence and the sharp-spectral filter, the residual velocity field5 

Denoting the LES velocity PDF by /u+V U t ),8 the mean and covariance with respect to the LES velocity PDF are defined, respectively, by + 00 

Note that we have made use of the fact that for a homogeneous flow with an isotropic filter (u U ) = 0. More generally, the conditional expected value of the residual velocity field will depend on the filter choice. 

Compared with the Reynolds stresses ( , + less is known concerning the statistical properties of the residual Reynolds stresses ( + U ). However, because they represent 

More precisely, the Fourier coefficients in (4.27) can be replaced by random variables with the following properties k. UK. = 0 and (UK.) = 0 for all k such that k kc. An energy-conserving scheme would also require that the expected value of the residual kinetic energy be the same for all choices of die random variable. The LES velocity PDF is a conditional PDF that can be defined in die usual manner by starting from die joint PDF for the discrete Fourier coefficients U . 

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The velocity field is assumed to be statistically homogeneous. Thus, die LES velocity PDF does not depend explicitly on x, only implicitly through U (x, t). 

Note that hv operates on the random field U(r, f) and (for fixed parameters V, x, and t) produces a real number. Thus, unlike the LES velocity PDF described above, the FDF is in fact a random variable (i.e., its value is different for each realization of the random field) defined on the ensemble of all realizations of the turbulent flow. In contrast, the LES velocity PDF is a true conditional PDF defined on the sub-ensemble of all realizations of the turbulent flow that have the same filtered velocity field. Hence, the filtering function enters into the definition of /u u(V U ) only through the specification of the members of the sub-ensemble. 

The resolved velocity U would then be found from an LES simulation, and the LES velocity PDF (defined in Section 4.2) would be written in terms of the unresolved velocity uL Alternatively, the filtered density function (FDF) approach can be used with a variant... 

As with the LES velocity PDF, a conditional PDF for the residual scalar field can be developed in terms of the LES composition PDF, denoted by / u OA lU, f).10 For a homogeneous scalar field with an isotropic filter, the conditional expected value of the scalar will have the property ( U,

transport equation can be derived for the residual scalar variance defined by11... 

In Section 4.2, the LES composition PDF was introduced to describe the effect of residual composition fluctuations on the chemical source term. As noted there, the LES composition PDF is a conditional PDF for the composition vector given that the filtered velocity and filtered compositions are equal to U and 0, respectively. The LES composition PDF is denoted by U, 0 x, /), and a closure model is required to describe it. 

Alternatively, an LES joint velocity, composition PDF can be defined where both (j> andU are random variables Aj 0 U 4 U >4 x, t). In either case, the sample space fields U and0 are assumed to be known. 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

Transported PDF methods combine an exact treatment of chemical reactions with a closure for the turbulence field. (Transported PDF methods can also be combined with LES.) They do so by solving a balance equation for the joint one-point, velocity, composition PDF wherein the chemical-reaction terms are in closed form. In this respect, transported PDF methods are similar to micromixing models. 

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. 

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. 

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. 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

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