PDF model

Falk, L. and Schaer, E., 2001. A PDF modelling of precipitation reactors. Chemical Engineering Science, 56, 2445-2458. 

V. Robin, A. Mura, M. Champion, and P. Plion 2006, A multi Dirac presumed PDF model for turbulent reactive flows. Combust. Sci. Technol. 178 1843-1870. 

In situ adaptive tabulation FHigher-order PDF models... 

Multi-environment presumed PDF models are generally not recommended for homogeneous flows with uniform mean gradients. Indeed, their proper formulation will require the existence of a mixture-fraction vector that, by definition, cannot generate a uniform mean scalar gradient in a homogeneous flow. 

Five examples of multi-environment presumed PDF models are summarized in Tables 5.1-5.5. For each case, the flow is assumed to be non-premixed with a mixture... 

Multi-environment presumed PDF models can also be easily extended to treat cases with more than two feed streams. For example, a four-environment model for a flow with three feed streams is shown in Fig. 5.24. For this flow, the mixture-fraction vector will have two components, 2 and 22- The micromixing functions should thus be selected to agree with the variance transport equations for both components. However, in comparison with multi-variable presumed PDF methods for the mixture-fraction vector (see Section 5.3), the implementation of multi-environment presumed PDF models in CFD calculations of chemical reactors with multiple feed streams is much simpler. 

The ability of multi-environment presumed PDF models to predict mean compositions in a turbulent reacting flow will depend on a number of factors. For example, their use with chemical kinetic schemes that are highly sensitive to the shape (and not just the low-order moments) of the joint composition PDF will be problematic for small Ne. Such... 

Note finally that, for any given value of the mixture fraction (i.e., f f), the multienvironment presumed PDF model discussed in this section will predict a unique value of 4>. In this sense, the multi-environment presumed PDF model provides a simple description of the conditional means (0 f) at Ve discrete values of f. An obvious extension of the method would thus be to develop a multi-environment conditional PDF to model the conditional joint composition PDF / (-i/d x, / ). We look at models based on this idea below. 

In a multi-environment conditional PDF model, it is assumed that the composition vector can be partitioned (as described in Section 5.3) into a reaction-progress vector y>rp and a mixture-fraction vector . The presumed conditional PDF for the reaction-progress vector then has the form 155... 

The multi-environment conditional PDF model thus offers a simple description of the effect of fluctuations about the conditional expected values on the chemical source term. 

The connection between the multi-environment conditional and unconditional PDF models can be made by noting that... 

For this reason, mixing in the conditional PDF model is orthogonal to mixing in the unconditional model. Thus, y in the conditional model need not be the same as in the unconditional model, where its value controls tiie mixture-fraction-variance decay rate. 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

As compared with the other closures discussed in this chapter, computation studies based on the presumed conditional PDF are relatively rare in the literature. This is most likely because of the difficulties of deriving and solving conditional moment equations such as (5.399). Nevertheless, for chemical systems that can exhibit multiple reaction branches for the same value of the mixture fraction,162 these methods may offer an attractive alternative to more complex models (such as transported PDF methods). Further research to extend multi-environment conditional PDF models to inhomogeneous flows should thus be pursued. 

Multi-environment presumed PDF models can be developed for the LES composition PDF using either the unconditional, (5.341), or conditional, (5.396), form. However, in order to simplify the discussion, here we will use the unconditional form to illustrate the steps needed to develop the model. The LES composition PDF can be modeled by163... 

It is now necessary to formulate transport equations for p and (s). However, by making the following substitutions, the same transport equations as are used in the multienvironment PDF model can be used for the multi-environment LES model ... 

We have used Fick s law of diffusion with separate molecular diffusivities for each species. However, most PDF models for molecular mixing do not include differential-diffusion effects. 

We shall see that the last term leads to a random noise term in die Lagrangian PDF model for die velocity. 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

Despite the obvious advantages of using a transported PDF model for , in practice it... 

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

The primary task in Lagrangian PDF modeling is thus to find appropriate functional forms for the drift (a, a >) and diffusion (Bj/u, Bj/, B j/, B ) coefficients. Fortunately, in many applications only a small sub-set of the coefficients will be non-zero. For example, for constant-density flows, au and Bf/f/ are independent of ( ), and both Bjand B u are... 

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. 

Lagrangian PDF modeling then consists of finding expressions for the coefficients on the right-hand sides of (6.172) and (6.173) that are consistent with the known behavior of the conditional fluxes. 

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... 

The transported PDF models discussed so far in this chapter involve the velocity and/or compositions as random variables. In order to include additional physics, other random variables such as acceleration, turbulence dissipation, scalar dissipation, etc., can be added. Examples of higher-order models developed to describe the turbulent velocity field can be found in Pope (2000), Pope (2002a), and Pope (2003). Here, we will limit our discussion to higher-order models that affect the scalar fields. 

In the joint velocity, composition PDF description, the user must supply an external model for the turbulence time scale r . Alternatively, one can develop a higher-order PDF model wherein the turbulence frequency > is treated as a random variable (Pope 2000). In these models, the instantaneous turbulence frequency is defined as... 

The DQMOM results for the IEM model can be compared with the multi-environment presumed PDF models in Section 5.10. In particular, (5.374) on p. 226 can be compared with (B.44), and (5.375) can be compared with (B.45). First, we can note that for the IEM model G = 0 and Gs = a. Likewise, y AT"1 + pnSa 4>)n) = 7 .,z and = Ban. Of the four models introduced in Tables 5.1-5.5, only the symmetric two-environment model in Table 5.2 has G = 0 and yM(an) + pnSa = 7Zan. However, because the spurious dissipation terms only ensure that the mixture-fraction variance is correctly predicted, the symmetric two-environment model does not have Gs = a and Mg 1 = Ban. Thus, the covariance matrix is not predicted correctly, as it would be if (B.43) were used. We can thus conclude that the multi-environment presumed PDF models are incomplete in the sense that they do not control as many of the moments as possible for a given choice of -/Ve. 

The results from (B.43) can be combined with the multi-environment presumed PDF models to define a new class of micromixing models that agree with the DQMOM result for inhomogeneous flows. Formally, (B.43) can be written as... 

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