Overview of the closure problem

In this section, we first introduce the standard form of the chemical source term for both elementary and non-elementary reactions. We then show how to transform the composition vector into reacting and conserved vectors based on the form of the reaction coefficient matrix. We conclude by looking at how the chemical source term is affected by Reynolds averaging, and define the chemical time scales based on the Jacobian of the chemical source term. 

As noted in Chapter 1, the chemical source term S(0) in (1.28) on p. 16 is the principal cause of closure difficulties when modeling turbulent reacting flows. The standard expression1 for the chemical source term begins with a set of I elementary reactions (Hill 1977) 2 

The chemical-source-term closures described in this chapter do not require that the chemical source term be derived from a set of elementary reactions as done here. Indeed, closure difficulties will result whenever S( / ) is a non-linear function of j . 

In elementary reactions, chemical elements are conserved. For example, in 2H2 +02t 2H20 

Based on (5.1), the chemical source term for the chemical species has the form5 

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In Chapter 4 the GPBE is derived, highlighting the closures that must be introduced for the passage from the microscale to the mesoscale model. This chapter also contains an overview of the mathematical steps needed to derive the transport equations for the moments of the NDF from the GPBE. The resulting moment-closure problem is also throughly discussed. 

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