Modeling of the Influence of Mixture Fraction Fluctuations on Burning Rate in Partially Premixed Turbulent Flames

Abstract

A common approach to modeling the influence of turbulent fluctuations in the mixture fraction f on the burning rate in a partially premixed flame consists of invoking a presumed Favre (mass-weighted) probability density function (PDF) . In the present work, two issues relevant to such an approach are addressed. First, averaging of a dependence q(f), where q is an arbitrary quantity, e.g., the laminar flame speed S_L, requires modeling of the canonical PDF P(f) if q is not proportional to the density . Second, because the shape of is not known a priori in a typical case, the presumed PDF approach can be a predictive tool only if the mean quantities are weakly sensitive to the PDF shape. To study the two issues, dependencies of and , computed for gasoline surrogate-air mixtures under elevated temperatures and pressures, associated with the conditions in a gasoline direct injection spark ignition engine, are averaged invoking either beta function or double-Dirac delta function Favre or canonical PDFs. Moreover, a simpler approach is proposed to evaluate . The approach consists of expanding in Taylor series with respect to , followed by averaging. The mean quantities and obtained for various Favre first and second moments using the aforementioned alternative methods are compared with each other and with and , respectively. The following conclusions are drawn. First, when averaging under conditions of the present study, the difference between the Favre and canonical beta-function PDFs may be disregarded for simplicity. Second, is sensitive to the shape of presumed PDF if the magnitude of turbulent fluctuations in the mixture fraction is sufficiently large. Third, if the magnitude of turbulent fluctuations in the mixture fraction is sufficiently low in order for the mean laminar flame speeds obtained invoking the beta-function and double-Dirac delta-function PDFs to be approximately equal to one another, then, can also be evaluated using the Taylor-expansion approach.

Keywords:

• Gasoline direct injection engine
• Mixture-fraction PDF
• Modeling
• Partially premixed turbulent combustion
• Presumed PDF

Notes

¹ The rate can also depend on spatial derivatives of the c(x, t) and f(x, t) fields (Bray et al., 2005), but this issue is beyond the scope of the present article.

² The computed S_L(f) curves were approximated in the entire range of , i.e., the approximated laminar flame speed did not vanish in inflammable mixture characterized by a low computed .

³ In our previous paper (Huang and Lipatnikov, 2012), the sensitivity was not stressed, because (i) the focus of discussion was placed on comparison of and in statistically near-stoichiometric mixtures; (ii) the magnitude of fluctuations in f was restricted by the flammability limits; and (iii) not only the BF and DDF PDFs, but also the R-DDF PDF were characterized by approximately the same third moments under conditions of that simulations. Nevertheless, the sensitivity is indicated in Figures 14 and 15 in the cited paper.

⁴ We tested also the third-order TE approach, i.e, (i) terms that involved were retained in Taylor series, (ii) was evaluated using the BF PDF (see the Appendix), (iii) an expression for was invoked similarly to Eq. (40) for and Eq. (41) for , and (iv) the obtained system of three linear equations was solved to determine , , and . However, such a complication of the numerical method improved agreement between the BF PDF and TE results only slightly.