Chemical source term conditional

In other closures for the chemical source term, a model for the conditional scalar dissipation rate (e

scalar Laplacian, the conditional scalar... 

Another conditional expectation that frequently occurs in closures for the chemical source term is the conditional mean of the composition variables given the mixturefraction. The latter, defined in Chapter 5, is an inert scalar formed by taking a linear combination of the components of 0 ... 

This, in turn, implies that the conditional chemical source term is closed when written in terms of the conditional means ... 

In developing closures for the chemical source term and the PDF transport equation, we will also come across conditional moments of the derivatives of a field conditioned on the value of the field. For example, in conditional-moment closures, we must provide a functional form for the scalar dissipation rate conditioned on the mixture fraction, i.e.,... 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix T into a vector c that has Nr reacting components cr and N conserved components cc.35 In the limit of equilibrium chemistry, the behavior of the Nr reacting scalars will be dominated by the transformed chemical source term S. 36 On the other hand, the behavior of the N conserved scalars will depend on the turbulent flow field and the inlet and initial conditions for the flow domain. However, they will be independent of the chemical reactions, which greatly simplifies the mathematical description. 

Note that the numerical simulation of the turbulent reacting flow is now greatly simplified. Indeed, the only partial-differential equation (PDE) that must be solved is (5.100) for the mixture-fraction vector, which involves no chemical source term Moreover, (5.151) is an initial-value problem that depends only on the inlet and initial conditions and is parameterized by the mixture-fraction vector it can thus be solved independently of (5.100), e.g., in a pre(post)-processing stage of the flow calculation. For a given value of , the reacting scalars can then be stored in a chemical lookup table, as illustrated in Fig. 5.10. 

Figure 5.21. Scatter plot of concentration in a turbulent reacting flow conditioned on the value of the mixture fraction. Although large fluctuations in the unconditional concentration are present, the conditional fluctuations are considerably smaller. In the limit where the conditional fluctuations are negligible, the chemical source term can be closed using the conditional scalar means. 

Extending (5.286) to conditional scalar moments of arbitrary order, the conditional chemical source term can be approximated by... 

For simple chemistry, a form for Q( x, t) can sometimes be found based on linear interpolation between two limiting cases. For example, for the one-step reaction discussed in Section 5.5, we have seen that the chemical source term can be rewritten in terms of a reaction-progress variable Y and the mixture fraction f. By taking the conditional expectation of (5.176) and applying (5.287), the chemical source term for the conditional reaction-progress variable can be found to be... 

In (5.297), the interpolation parameter is defined separately for each component. Note, however, that unlike the earlier examples, there is no guarantee that the interpolation parameters will be bounded between zero and one. For example, the equilibrium concentration of intermediate species may be negligible despite the fact that these species can be abundant in flows dominated by finite-rate chemistry. Thus, although (5.297) provides a convenient closure for the chemical source term, it is by no means guaranteed to produce accurate predictions A more reliable method for determining the conditional moments is the formulation of a transport equation that depends explicitly on turbulent transport and chemical reactions. We will look at this method for both homogeneous and inhomogeneous flows below. 

This boundary condition does not ensure that the unconditional means will be conserved if the chemical source term is set to zero (or if the flow is non-reacting with non-zero initial conditions Q( 0) 0). Indeed, as shown in the next section, the mean values will only be conserved if the conditional scalar dissipation rate is chosen to be exactly consistent with the mixture-fraction PDF. An alternative boundary condition can be formulated by requiring that the first term on the right-hand side of (5.299) (i.e., the diffusive term) has zero expected value with respect to the mixture-fraction PDF. However, it is not clear how this global condition can be easily implemented in the solution procedure for (5.299). 

For simple chemistry, we have seen in Section 5.5 that limiting cases of general interest exist that can be described by a single reaction-progress variable, in addition to the mixture fraction.131 For these flows, the chemical source term can be closed by assuming a form for the joint PDF of the reaction-progress variable Y and the mixture fraction . In general, it is easiest to decompose the joint PDF into the product of the conditional PDF of Y and the mixture-fraction PDF 132... 

Unlike for the mixture fraction, the initial and inlet conditions for the mean and variance of Y will be zero. Thus, the chemical source term will be responsible for the generation of non-zero values of the mean and variance of the reaction-progress variable inside the reactor. Appealing again to the assumption of independence, the mean reaction-progress variable is given by... 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

Following the example of (5.307), the model equations are written in terms of the mixture-fraction-PDF-weighted conditional means. The integral of (5.399) over mixture-fraction space will eliminate the first two terms on the right-hand side, leaving only the mean chemical source term as required. 

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. 

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. 

The composition PDF thus evolves by convective transport in real space due to the mean velocity (macromixing), by convective transport in real space due to the scalar-conditioned velocity fluctuations (mesomixing), and by transport in composition space due to molecular mixing (micromixing) and chemical reactions. Note that any of the molecular mixing models to be discussed in Section 6.6 can be used to close the micromixing term. The chemical source term is closed thus, only the mesomixing term requires a new model. 

Because the chemical source term is treated exactly in transported PDF methods, the conditional reaction/diffusion term,... 

Based on the above examples, we can conclude that while localness is a desirable property, it is not sufficient for ensuring physically realistic predictions. Indeed, a key ingredient that is missing in all mixing models described thus far (except the FP and EMST70 models) is a description of the conditional joint scalar dissipation rates (e ) and their dependence on the chemical source term. For example, from the theory of premixed turbulent flames, we can expect that (eY F, f) will be strongly dependent on the chemical... 

Note that the vector functions go and gi will normally be time-dependent, but can be found from the conditional moments (01 %). In the transported PDF context, the latter can be computed directly from the joint composition PDF so that g0 and gi will be well defined functions.110 The FP model in this limit is thus equivalent to a transported PDF extension of the conditional-moment closure (CMC) discussed in Section 5.8.111 The FP model (including the chemical source term S(0, f)) becomes... 

In the next fractional time step, the change due to the chemical source term is found separately for each notional particle using 0l" Xf + At) as the initial condition ... 

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 ... 

A particularly simple method of approximating the chemical source term co(Y) is reaction elimination all one does is delete reactions which are numerically insignificant under the current reaction conditions. As shown by Bhattacharjee et al. (2003), at any set of single point reaction condition(s), one can rigorously identify the smallest possible set of reactions which reproduces the rate at which species are made or consumed chemically, and the rate at which heat is released due to chemical reactions, cumulatively >( Y), to within a user-specified tolerance vector tols , Eq. (16). Mathematically, the process of finding the smallest possible model is a constrained interval optimization... 

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