A Simple Model for Evaluating Conditioned Velocities in Premixed Turbulent Flames

Abstract

A simple model is proposed to evaluate (a) the divergence of velocity vector conditioned on unburned mixture, and (b) the vector component normal to the mean flame brush in the flamelet regime of premixed turbulent combustion. The model involves a single constant and does not invoke an extra balance equation. To perform the first test of the model, six flames stabilized in impinging jets and experimentally investigated by 4 research groups were numerically simulated. In the computations, (a) approximations of the measured axial profiles of the mean combustion progress variable were invoked, (b) the well-known (Bray et al., 1998, and 2000) statistically steady and 1-dimensional Favre-averaged continuity and Euler equations were numerically integrated in order to approximate the measured axial profiles of the mean axial velocity, and, then, (c) the approximations were utilized in order to evaluate conditioned velocities and turbulent scalar flux using the proposed model supplemented with the BML approach and balance equation for the Favre-averaged combustion progress variable. The obtained agreement between the measured and computed axial profiles of the conditioned axial velocities or axial turbulent scalar flux was encouraging, thus, indicating that the proposed simple model is promising. Since the correlation between fluctuations of velocity and unity normal vectors, conditioned to flamelet surface, plays a key role in the model, the encouraging test results call for studying this correlation in future DNS. Moreover, further research into the difference in velocity conditioned on unburned mixture and velocity conditioned on the unburned side of flamelets is necessary for improving the model at the leading edge of a turbulent flame brush.

Keywords:

• Conditioned velocities
• Countergradient transport
• Modeling
• Premixed turbulent combustion
• Stagnation flames

ACKNOWLEDGMENTS

The first author (VS) was supported by ONERA and the second author (AL) was supported by the Swedish Research Council (VR), the Swedish Energy Agency, and by the Combustion Engine Research Center (CERC). We also gratefully acknowledge the hospitality and support of the Nordic Institute for Theoretical Physics and, in particular, of Axel Brandenburg during the NORDITA “Turbulent Combustion” program where a part of this work was done.

Notes

^a d is the distance between the jet exit and the wall (see Figure 2).

^b U is the mean axial flow velocity in the jet exit.

^c Φ is the equivalence ratio.

^d Values of S _L for CH₄-air mixtures were taken from the paper by Bosschaart and de Goey (2004).

^e This value was reported by Bray et al. (2000, see footnote on p. 645 in the cited paper).

^f This value was reported by Cheng and Shepherd (1991).

^g The main difference between the two flames studied by Li et al. (1994) was the diameters (h = 4 and 6 mm) of holes in grids used to generate turbulence.

The problem of modeling these terms is beyond the scope of the present paper and the interested reader is referred to review papers by Lipatnikov and Chomiak (2002, 2010) and by Veynante and Vervisch (2002).