### 7.9.1. Reaction Progress

A single progress variable, , is used to describe the progress of the global reaction:

| (7–39)

The composition of the fluid is determined by blending the compositions of the non-reacted state (fresh gases) and the reacted state (burned gases), where corresponds to fresh materials and corresponds to fully reacted materials.

In turbulent flow, a bimodal distribution of is assumed. At any given time and position in space the fluid is considered to be either fresh materials or fully reacted. This assumption is justified if the chemical reaction is fast compared to the integral turbulent time scales of the flow.

Then, the averaged reaction progress variable, , is the probability for the instantaneous state of the fluid being reacted. The mean species composition of the fluid is computed according to:

| (7–40)

For example, if , then the fluid at the given position will be fully reacted during 60% of time and non-reacted during the remaining 40% of time.

The reaction progress variable is computed by solving a transport equation:

| (7–41)

The default value of the turbulent Schmidt number for the reaction progress variable is .

In the limits of pure fuel and pure oxidizer, the reaction progress is not well defined because burnt and unburnt conditions correspond to the same physical state:

| (7–42)

| (7–43)

This poses the issue of which boundary value to specify for when the mixture is either pure fuel or pure oxidizer. Even though different values for correspond to the same mixture composition, it is still important to impose the proper value on the boundary, because it controls the combustion regime in the domain after mixing has occurred:

- corresponds with premixed combustion

- corresponds with non-premixed combustion (diffusion flame)

In most cases, is appropriate for fuel inlets (in fact, if for fuel the Flamelet model could be used and not solve for at all). For oxidizer inlets, the appropriate boundary value depends on the mixing process in the domain:

- for oxidizer premixing with fuel (e.g., primary air)

- for oxidizer mixing with products (e.g., secondary air)

However, which case applies may not be known prior to the simulation. In case of a flow split, it may even be that both cases occur for the same oxidizer inlet. In this situation, the artificial distinction between “burnt air” () and “fresh air” () may cause unphysical behavior such as even fuel being generated by mixing of products with fresh air. For example, mixing “fresh air” (, ) with products (, ) would result in a mixture that is only partially burnt (, ). The reaction progress of the resulting mixture equals the fraction of products , equivalent to fuel being re-established. This is unphysical. The correct behavior would be obtained by mixing products with “burnt air” (, ).

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