Modeling the nonlinear aeroacoustic response of a harmonically forced side branch aperture under turbulent grazing flow

Steady-state statistics, emergent patterns and intermittent energy transfer in a ring of oscillators

“…Experimental and numerical studies highlight also the role of the nonlinear saturation of the aeroacoustic response of a shear layer [38,39]: When the acoustic forcing amplitude is increased, the response of the shear layer, in terms of its kinetic energy or its acoustic energy production at the fundamental frequency of excitation, decreases. A theoretical model of this nonlinear shear layer response, based on Howe's formulation, was derived and validated against experiments in [40]. The same saturation mechanism leads to self-excited aeroacoustic instabilities, which occur, for example, when we whistle.…”

“…The sign before ϑ is positive because the coupling is amplifying if λ < 0 and we assume that as in Ref. [40], the constructive aeroacoustic coupling saturates at high amplitudes (monotonic decay of constructive feedback strength). For simplicity, this work focuses on situations where the nonlinear aeroacoustic coupling can be considered as a small perturbation of the thermoacoustic oscillations in the cans, so that the saturation associated with the thermoacoustic instability is significantly larger than the saturation of the aeroacoustic coupling: κ ϑ.…”

“…Taking the time derivative of Eqs. (39) and (40) and substituting Eq. ( 41) into the resulting expressions yields…”

Coupling-induced instability in a ring of thermoacoustic oscillators

“…To arrive at an expression for Z, we use the Rayleigh conductivity, which is defined as follows [38,40,46]: KR=−sρΦfalse^false[pfalse^false], where Φfalse^ is the outward facing coherent volume flux through the aperture, resulting from oscillatory motion of the vortex sheet, and false[pfalse^false] is the acoustic pressure difference across the aperture. By equating the coherent volume flux Φfalse^ to the acoustic volume flux through the aperture Aau^a, where Aa=WB is the aperture area and u^a is the transverse acoustic velocity of the fluid in the aperture, we obtain a relation between KR and the specific acoustic impedance Zs=Z/ρc [42–44]: Zs=false[pfalse^false]ρcu^a=−s…”

“…Assuming (a) and (b) describe the same point (Imfalse(KRfalse)>0 or Refalse(Zfalse)<0 both imply amplification of the sound field by the mean flow in the aperture [44]), this implies U/Utot≈0.42 for the experiments of [45]. We also mention the study in [44], where a Rayleigh conductivity model was calibrated to experimental results to obtain a predictive model of the acoustic impedance of a side branch aperture. After calibration, the value of U/Utot obtained therein is within 1 per cent of 0.5 (U corresponds to U− in their notation).…”