In a previous lecture we discussed turbulence modelling, in which the aim was to remove the
‘small scale’ detail of turbulence and replace it with a (cheaper) turbulence model. This is
done by an averaging process. A similar approach is used in combustion modelling to eliminate
the small scale turbulent effects and to derive models for the combustion processes. Since the
density of the fluid is variable we introduce a density-weighted average called a Favre
average. As usual we split the flow up into mean and fluctuating '' components
![]() | (VII.5) |
![]() | (VII.6) |
![]() | (VII.7) |
We can also apply Favre averaging to equation (VII.4). After some manipulation we arrive at the equation
![]() | (VII.8) |
- | ![]() ![]() ![]() | representing turbulent transport, and | |||||
![]() ![]() | the mean chemical source term |
Devised by Spalding (1971), this is based on the assumption that the turbulent cascade mixing process controls the chemical reactions - i.e. that mixing of the reactants is the rate-determining step in the process. This can be used to derive a simple model for the chemical source term. However this model requires detailed tuning of coefficients, an unsatisfactory arrangement, and is not physically very accurate. It has been formulated primarily for premixed combustion, although it is possible to develop formulations for non-premixed combustion.
Turbulence can be discussed statistically, as a quasi-random flow with particular
statistical properties. These properties can be described in terms of correlations
between flows at different points, or the same point at different times. The basic
quantity involved is the PDF, or Probability Density Function. Simple engineering
models such as the k - model ignore this, providing information only about
the mean levels of turbulence. However it is possible to model turbulence in a
statistical manner, providing information about these turbulent correlations as a
PDF transport equation. Such models are computationally costly but can be very
accurate.
Extending this to cover combustion involves providing a transport equation for the so-called joint pdf of velocity and reactive scalars. This is denoted