A new multi-scale mixing model for turbulent reacting flows
2003
In the probability density function (PDF) description of a turbulent reacting flow, the local
temperature and species concentration are replaced by a high-dimensional joint probability
density function that describes the distribution of states in the fluid. The PDF has the
great advantage of rendering the chemical reaction source terms closed, independent of their
complexity. However, molecular mixing, which involves two-point information, must be modeled.
Indeed, the qualitative shape of the PDF is sensitive to this modeling, hence the reliability
of the model to predict even the closed chemical source terms rests heavily on the mixing
model.
Spectral models contain two-point
statistical information of the velocity and scalar fields that can accurately capture
all of the spectral dynamics of the scalars (enthalpy and species mass fractions).
A new closure called `multi-scale mixing model'
is presented for the mixing terms based on a spectral representation
of the scalar field.
Such a closure is developed in two levels. In level A,
the eddy damped quasi normal Markovian (EDQNM)
model, extended to multiple scalars by Ulitsky and Collins (Journal of Fluid Mechanics, 412, 2000),
has been applied to the study of mixing of differential diffusing scalars
and completely
validated by direct numerical simulations (DNS).
In level B, the `multi-scale model' of mixing is then developed
based on this EDQNM closure.
This model is implemented as an ensemble of stochastic particles, each
carrying scalar concentrations distributed across different wavenumbers. Scalar exchanges
within a given particle represent ``transfer' while scalar exchanges between particles
represent ``mixing.'
The model
correctly predicts the evolution of an
initial double delta function PDF (unmixed condition) into a
self-similar
Gaussian as seen in DNS by Eswaran and Pope (Physics of Fluids, 31, 1988).
Comparisons of the model with DNS are in good agreement and
extensions to multiple scalar mixing are also presented.
To bring in the effect of chemical reactions,
a new EDQNM model for a bimolecular reaction
is developed (level A). This model is validated
with DNS for a bimolecular reaction for unmixed initial conditions.
A widely used PDF mixing model, Interaction by Exchange with Mean (Dopazo, Physics of Fluids,18, 1975) model fails even
to qualitatively capture the product-reactant correlations, an essential
statistic for multi-step chemical reactions, while EDQNM, due to the fact that it retains
spectral information, is able to do a better job at predicting this statistic.
Work has been initiated to include the effect of reaction into the
multi-scale mixing model described earlier.
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