Modeling the Response of Turbulent Flames to Harmonic Forcing

ABSTRACT

This article analyzes the response of a turbulent, premixed flame to harmonic forcing. This problem has been worked extensively for laminar flames, and the key parameters influencing the flame transfer function are well understood. For turbulent flames, several prior studies have utilized a “quasi-laminar” approach, by utilizing the time-averaged flame position and ensemble-averaged disturbance field, as inputs to what is otherwise identical to the laminar problem. More generally, the manner in which turbulent flames respond to harmonic disturbances is not amenable to analytical solutions because of the nonlinear interactions between stochastic flow disturbances and harmonic flame wrinkling. We utilize a turbulent burning velocity closure proposed by Shin and Lieuwen (2013), who showed that the ensemble-averaged turbulent burning rate for a harmonically forced flame is proportional to the ensemble-averaged flame curvature. Shin and Lieuwen (2013) previously used it to analyze the ensemble-averaged space-time flame wrinkle characteristics. Here, we extend these results to analyze the spatial variation of ensemble-averaged flame surface area and burning rate and then compare these results to computations. These results show that, for low stochastic forcing amplitudes, wrinkling of the front exerts quantitative differences between those predicted by a quasi-laminar and the actual flame response (e.g., reducing peak values of the flame transfer function and eliminating nodes), but does not change the key qualitative features. While this result needs to be considered for strongly turbulent flames, it does suggest why good agreement has been observed between quasi-laminar approaches and experimental data for harmonically excited, turbulent flames. Two results for model problems showing the linearized flame transfer functions are also presented, which explicitly demonstrate qualitative turbulence effects on harmonically excited flames.

KEYWORDS:

• Level-set
• Quasi-laminar
• Transfer function
• Turbulent heat release

Funding

This work has been partially sponsored by the NASA Aeronautics Scholarship Program through grant number NNX14AE98H, and the National Science Foundation contract CBET-1235779 (contract monitor Professor Ruey-Hung Chen).

Nomenclature

Table

Notes

¹ As noted in Humphrey et al. (2014), multiple definitions for the spatially integrated heat release exist, depending upon one’s assumptions of the potentially oscillating integration limits. We will assume here that flames are confined and spread to the wall and so the transverse integration limits are fixed, implying that the axial integration limits oscillate.

² Note that an analogous approach is sometimes used in the hydrodynamic stability literature, where the time averaged velocity profile of a turbulent flow is used as an input to a stability calculation to determine the growth rate of a harmonic space/time disturbance; see a discussion of this approach in, e.g., Barkley (2006), Meliga et al. (2012), and Mettot et al. (2014).

³ Note, however, that the numerical results and non-linear analytical integration (see Figure 7) show that the FDF gain can increase with turbulence if the mean flame speed and flame slope changes with downstream distance.

Additional information

Funding

This work has been partially sponsored by the NASA Aeronautics Scholarship Program through grant number NNX14AE98H, and the National Science Foundation contract CBET-1235779 (contract monitor Professor Ruey-Hung Chen).