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Presumed Joint-PDF Modelling for Turbulent Stratified Flames

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Abstract

Turbulence–chemistry interaction models such as the conditional moment closure and various flamelet models require a presumed Probability Density Function (PDF) model of the conditioning variables. In turbulent stratified flames, a joint-PDF model for a reaction progress variable and the mixture fraction is required. The joint-PDF is often modelled by two beta functions using the first and second moments of the two conditioning variables. In this work, the performance of a joint-PDF model based on the flamelet PDF approach is compared to the double beta-PDF model. The conditional PDF of the reaction progress variable conditioned on mixture fraction is modelled with the flamelet PDF. Unlike the beta PDF, the flamelet functional form for the conditional PDF varies with the local mixture fraction. The two PDF models are coupled with a two-dimensional premixed flamelet-generated manifold chemistry model. Two stratified flames of the Cambridge–Sandia bluff-body burner are simulated using a Reynolds-Averaged Navier–Stokes (RANS) approach. The results indicate that the flamelet PDF model has a superior, albeit marginal, predictions of the temperature, major and minor species mass fractions.

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Abbreviations

a :

Strain rate

\(c^*\) :

Sample space variable for normalized reaction progress variable

\(C_{\delta _0}\), \(C_{\nabla }\), \(C_{\delta _1}\), \(c_1\) and \(c_2\) :

Flamelet PDF model constants

\(C_{\varepsilon _z}\) :

Mixture fraction dissipation rate model constant

\(C_3\) and \(C_4\) :

Progress variable dissipation rate model constants

f :

Interior distribution in the flamelet PDF

H :

Heaviside function

\(\widetilde{k}\) :

Favre-averaged turbulent kinetic energy

Ka :

Karlovitz number

P :

Probability density function

\(Sc_t\) :

Turbulent Schmidt number

\(S_L\) :

Laminar flame speed

T :

Temperature

\(u_i\) :

Velocity field

\(Y_c\) :

Non-normalized reaction progress variable

\(\widetilde{Y}_c\) :

Favre-averaged non-normalized reaction progress variable

\(\widetilde{Y}_{c_v}\) :

Non-normalized reaction progress variable variance

\(Y_k\) :

Mass fraction of the kth-species

\(z^*\) :

Sample space variable for mixture fraction

\(\widetilde{z}\) :

Favre-averaged mixture fraction

\(\widetilde{z}_v\) :

Mixture fraction variance

\(\beta _\varepsilon\) :

Progress variable dissipation rate model constant

\(\delta\) :

Dirac delta function

\(\delta _L\) :

Laminar flame thickness

\(\delta _z\) :

Zeldovitch thickness

\(\varGamma\) :

Gamma function

\(\widetilde{\varepsilon }\) :

Favre-averaged turbulent dissipation rate

\(\widetilde{\varepsilon }_z\) :

Favre-averaged mixture fraction dissipation rate

\(\widetilde{\varepsilon }_{Y_c}\) :

Progress variable dissipation rate

\(\eta\) :

Kolmogrov length scale

\(\mu _t\) :

Turbulent viscosity

\(\rho\) :

Density

\(\varSigma (c^*)\) :

Conditional flame surface density

\(\tau\) :

Heat release parameter

\(\phi\) :

Equivalence ratio

\(\dot{\omega }_k\) :

Production rate of the kth-species

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Acknowledgements

The authors much appreciate Prof. Simone Hochgreb for making the Cambridge–Sandia burner experimental data available to the combustion community.

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  1. Aerospace Engineering Department, Sharif University of Technology, Tehran, Iran

    Mohammadreza Ghadimi, Hassan Atayizadeh & M. Mahdi Salehi

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  1. Mohammadreza Ghadimi
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  3. M. Mahdi Salehi
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Correspondence to M. Mahdi Salehi.

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Ghadimi, M., Atayizadeh, H. & Salehi, M.M. Presumed Joint-PDF Modelling for Turbulent Stratified Flames. Flow Turbulence Combust 107, 405–439 (2021). https://doi.org/10.1007/s10494-021-00241-6

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