Acoustic radiation from weakly wrinkled premixed flames

This paper describes a theoretical analysis of acoustic radiation from weakly wrinkled (i.e., u′/SL<1u′/SL<1) premixed flames. Specifically, it determines the transfer function relating the spectrum of the acoustic pressure oscillations, P′(ω)P′(ω), to that of the turbulent velocity fluctuations in the approach flow, U′(ω)U′(ω). In the weakly wrinkled limit, this transfer function is local in frequency space; i.e., velocity fluctuations at a frequency ω distort the flame and generate sound at the same frequency. This transfer function primarily depends upon the flame Strouhal number St (based on mean flow velocity and flame length) and the correlation length, Λ  , of the flow fluctuations. For cases where the ratio of the correlation length and duct radius Λ/a≫1Λ/a≫1, the acoustic pressure and turbulent velocity power spectra are related by P′(ω)∼ω2U′(ω)P′(ω)∼ω2U′(ω) and P′(ω)∼U′(ω)P′(ω)∼U′(ω) for St≪1St≪1 and St≫1St≫1, respectively. For cases where Λ/a≪1Λ/a≪1, the transfer functions take the form P′(ω)∼ω2(Λ/a)2U′(ω)P′(ω)∼ω2(Λ/a)2U′(ω) and P′(ω)∼ω2(Λ/a)2(Ψ−Δln(Λ/a))U′(ω)P′(ω)∼ω2(Λ/a)2(Ψ−Δln(Λ/a))U′(ω) for St≪1St≪1 and St≫1St≫1, respectively, where Ψ and Δ are constants. The latter result demonstrates that this transfer function does not exhibit a simple power law relationship in the high frequency region of the spectra. The simultaneous dependence of this pressure–velocity transfer function upon the Strouhal number and correlation length suggests a mechanism for the experimentally observed maximum in acoustic spectra and provides some insight into the controversy in the literature over how this peak should scale with the flame Strouhal number.