Analysis of low-frequency wave scattering by turbulent premixed flame

Abstract

Theoretical investigation of acoustic wave interactions with turbulent premixed flames was conducted to evaluate the acoustic energy amplification and/or damping due to the interaction of low-frequency acoustic waves with turbulent flames in three-dimensional space. Such amplified or damped acoustic energy is either coherent or incoherent as wrinkled flames cause coherent energy of a monochromatic acoustic wave to be damped into incoherent energy of spatially diffused and spectrally broadened acoustic waves. Small perturbation method (SPM) up to the second order was utilized to analyse net coherent and incoherent acoustic energies of the reflected and transmitted waves scattered from a weakly wrinkled turbulent flame surface in random motion. General formulations for net coherent and incoherent energy budget of the scattered fields were derived that can be applied to any type of flame height statistics. Production and/or damping of acoustic energy scattered from a turbulent flame is attributed to two effects: one is the acoustic velocity jump due to flame's unsteady heat release and the other is the flame's wrinkling due to its unsteady motion. Dimensionless parameters that govern net acoustic energy budget were derived in case of Gaussian statistics of flame surface behaviour: the r.m.s. and correlation length of flame height, the frequency ratio of the incidence frequency to the flame's correlation frequency, the time ratio of the flame's diffusion to correlation time and the incidence angle. The results of the scattered acoustic energy budget showed that noticeable amplification of acoustic energy was obtained either for a small frequency ratio (≪1) at the critical incidence angle or for a large frequency ratio and time ratio (≫1), while damping was obtained for a small frequency ratio at off-critical incidence angles. The relative importance of unsteady heat release (the jump effect) and unsteady motion (the wrinkling effect) to net acoustic energy is controlled mainly by the frequency ratio: The unsteady heat release effect dominates the wrinkling effect at a large frequency ratio, and vice versa at a small frequency ratio. The energy transfer from coherent to incoherent energy is due to flame surface wrinkling and is enhanced with the square of the flame's r.m.s. height.