Numerical Methods for Reacting Flows

Numerical computations of reacting flows are often difficult owing to the different time-scales involved and the highly non-linear dependence of the reaction rate on concentrations and temperature. The solution of the species concentration equations in combination with the momentum and the enthalpy equation generally requires an iterative procedure such as the one outlined in Section 1.3.4. A rough sketch of the numerical structure of a stationary reacting-flow problem is given as 

Methods based on the partitioning of a reaction system into fast and slow components have been proposed by several authors [158-160], A key assumption made in this context is the separation of the space of concentration variables into two orthogonal subspaces and Qf spanned by the slow and fast reactions. With this assumption the time variation of the species concentrations is given as 

The notation is such that (Qs,j) , (Qf,j) denotes the ith component ofthejth basis vector in the subspace of slow and fast reactions, respectively. The corresponding expansion coefficients are (y )j and (yf )j, respectively, and are expressed by the reaction rates via 

If the time-scale of the fast reactions is much shorter than that of the slow reactions, it can be assumed that the former are completed at an initial stage of the latter. Mathematically, this assumption reads 

(122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161], 

The solution of the species concentration equations in combination with the momentum and the enthalpy equation generally requires an iterative procedure. A rough sketch of the numerical structure of a stationary reacting-flow problem is given as 

Apart from the coupling of chemical kinetics to the transport equations, the chemical-reaction dynamics itself may pose numerical challenges when a number of different reactions are superposed. In such a case the rate of disappearance of a chemical species i can be written as 

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