Scalar vectors conserved-variable

Figure 5.5. The conserved-scalar vector ipc can be partitioned by a linear transformation into two parts

Also, by applying the linear transformation to (5.65), we can see that the conserved-variable scalar vector obeys a transformed scalar transport equation of the form... 

Thus, the non-stationary turbulent reacting flow will be completely described by the first (Nr +N m) components of turbulent reacting flow, c(0) can be replaced by any of the inlet composition vectors c(l) for any /el,..., A m. For this case, if N m = Nin, then = Nin - 1 and thus one less conserved-variable scalar will be required to describe completely the stationary turbulent reacting flow. 

The interest in reformulating the conserved-variable scalars in terms of the mixture-fraction vector lies in the fact that relatively simple forms for the mixture-fraction PDF can be employed to describe the reacting scalars. However, if < /Vmf, then the incentive is greatly diminished since more mixture-fraction-component transport equations (Nmf) would have to be solved than conserved-variable-scalar transport equations (/V, << ). We will thus assume that N m = Nmf and seek to define the mixture-fraction vector only for this case. Nonetheless, in order for the mixture-fraction PDF method to be applicable to the reacting scalars, they must form a linear mixture defined in terms of the components of the mixture-fraction vector. In some cases, the existence of linear mixtures is evident from the initial/inlet conditions however, this need not always be the case. Thus, in this section, a general method for defining the mixture-fraction vector in terms of a linear-mixture basis for arbitrary initial/inlet conditions is developed. 

Note that the reaction-progress vector in the first column is non-zero. Thus, as we suspected, the mixture-fraction basis is not a linear-mixture basis. The same conclusion will be drawn for all other mixture-fraction bases found starting from (5.118). For these initial and inlet conditions, a two-component mixture-fraction vector can be found however, it is of no practical interest since the number of conserved-variable scalars is equal to Nq,m = 1 (k e 0, 1, 2). In conclusion, although the mixture fraction can be defined for the... 

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

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