Applicability of extrapolation relations for curvature and stretch rate dependences of displacement speed for statistically planar turbulent premixed flames

Abstract

A three-dimensional Direct Numerical Simulation (DNS) database of statistically planar turbulent premixed flames covering the wrinkled/corrugated flamelets and the thin reaction zones regimes of premixed turbulent combustion has been utilised to assess the performances of the extrapolation relations, which approximate the stretch rate and curvature dependences of density-weighted displacement speed $S_d^{\ast }$. It has been found that $S_d^{\ast }$ remains negatively correlated with curvature and exhibits a significantly non-linear relation with stretch rate for all cases considered here. As a result of these behaviours, an extrapolation relation, which assumes a linear stretch rate dependence of density-weighted displacement speed has been found to be inadequate. However, an alternative extrapolation relation which assumes a linear curvature dependence of $S_d^{\ast }$ has been found to be more successful in capturing local behaviour of the density-weighted displacement speed because it allows for a non-linear stretch rate dependence of $S_d^{\ast }$. The extrapolation relations, which express $S_d^{\ast }$ as non-linear functions of either curvature or stretch rate, have been found to perform better than the linear extrapolation relations. The Markstein length $L_M$ for all the extrapolation relations except for the one, which assumes a linear stretch rate dependence of $S_d^{\ast }$, has been found to be in good agreement with an analytical expression for $L_M$. An extrapolation relation which involves solving a non-linear equation in terms of stretch rate has been found to be sensitive to the initial guess and an alternative high-order polynomial-based extrapolation relation gives rise to overshoots and undershoots and thus the goodness of fit of these relations suffers for high turbulence intensities. An alternative extrapolation relation has been proposed in this analysis, which explicitly accounts for the non-linear curvature dependence of the combined reaction and normal diffusion components of $S_d^{\ast }$ and satisfactorily captures the statistical behaviours of $S_d^{\ast }$ for all cases considered here.

Keywords:

• density-weighted displacement speed
• curvature
• stretch rate
• extrapolation relation
• Markstein length

Acknowledgements

The authors are grateful to EPSRC (grant: EP/P022286/1) for financial support. The computational support was provided by ARCHER (grant: EP/R029369/1 and EP/K025163/1), CIRRUS, Leibniz Supercomputing Centre (grant: pn69ga), and HPC facility at Newcastle University (ROCKET). The authors are grateful to Prof. H. G. Im for the detailed chemistry DNS database.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are grateful to Engineering and Physical Sciences Research Council – EPSRC [grant number EP/P022286/1] for financial support. The computational support was provided by ARCHER [grant numbers EP/R029369/1 and EP/K025163/1], CIRRUS, Leibniz Supercomputing Centre [grant number pn69ga], and HPC facility at Newcastle University [ROCKET].