Helmholtz Resonator Impedance Model, Part 1: Nonlinear Behavior

Abstract

A one-dimensional preliminary design impedance prediction model is derived for Helmholtz resonators constructed with circular ori ces. The model addresses only the effects of resonator geometry and incident sound-pressure amplitude. The effects of grazing ow are not included. An experimental program was conducted to calibrate unknown empirical parameters derived in the model. The model was shown to predict, reasonably accurately, impedance data published in the open literature for resonators constructed with single and multiple ori ces over a wide range of sound-pressure amplitudes and frequencies. The model provides a rapid means of calculating resonator impedance as a function of resonator geometry, incident sound-pressure amplitude, and frequency. Nomenclaturece inertia length de ned by Eq. (19) d N = multiple-ori ce diameter de ned by Eq. (34) d o = single-ori ce diameter F = sound frequency H = ori ce nonlinear inertial length de ned by Eq. (20) and Fig. 1 K ac = acoustic viscous loss parameter de ned by Eq. (22) K ss = steady-state viscous loss parameter de ned by Eq. (21) k = sound wave number !=c 0 L c = cavity depth N = number of ori ces P c = peak cavity sound pressure de ned by Eq. (7) P pk = p 2P 0 P 0 = peak driving sound pressure acting on control volume upper surface S o R = real part of resonator impedance R fd = steady-state fully developed laminar resistive loss de ned by Eq. (16) R L = linear resistive loss de ned by Eq. (15) r o = ori ce radius S BL = ori ce boundary-layer area, S BL D S o ¡ S inv de ned in Fig. 1 S c= cavity cross-sectionalarea S inv = ori ce inviscid area de ned in Fig. 1 S N = multiple ori ce center-to-center spacing S o = ori ce area S w = ori ce wetted area t = time u BL = acoustic velocity passing through S BL de ned in Fig. 1 u inv = acoustic velocity passing through S inv de ned in Fig. 1 u o = ori ce peak acoustic velocity hu o i rms = rms amplitude averaged over each half-cycle V non = nondimensional acoustic particle velocity de ned by Eq. (23) X = imaginary part of resonator impedance ± = ori ce instantaneous boundary-layer thickness µ p = phase angle between P 0 and u o de ned by Eq. (5) · vis = ori ce viscous loss parameter de ned by Eq. (10) = sound wavelength ¹ = uid viscosity º = uid kinematic viscosity ½ 0 = uid density ¾= ori ce open area ratio, S o =S c ¿ = ori ce thickness ¿ w = acoustic shear stress acting on ori ce wetted area de ned by Eq. (8) ! = sound radian frequency (D 2¼ F) Subscripts c = cavity L = linear NL = nonlinear res = resonance

Nonlinear asymptotic impedance model for a Helmholtz resonator liner

“…The reactance shows a good comparison across the range of frequencies. The same comparison was done with other experimental configurations from [3] and a reasonably good agreement is found for the near resonance frequencies.…”

“…The Hersh et al model [3] to predict the impedance is based on the experimental calibration of empirical parameters that were derived in the formulation. They introduced six assumptions, mostly inspired from measurements, to model the nonlinear terms.…”

“…In these examples, the nonlinear corrections of the impedance are based on physically inspired modelling assumptions, but otherwise do not aim to solve the equations of the nonlinear resonator [21,12,3]. In contrast, the properties of the Helmholtz resonator have been obtained from the full equations in [6][7][8][9][10]16,17] but these are all fully CFD, DNS or LES simulations which do not give information for the simpler models.…”

Determination of Acoustic Impedance for Helmholtz Resonators Through Incompressible Unsteady Flow Simulations

“…Complementary computations have been run for a different Helmholtz resonator geometry, based on the case studied by Hersh et al [14] (see Fig. 12a in the referred work), to see if the overprediction in resistance is also present.…”

“…For the compressible simulations, in the case of the full resonator configuration, the inlet boundary condition is A f given by Eq. (14). Finally, exchanging the backing cavity with a nonreflecting outlet (Z ρc), the amplitude of the F r wave has to be corrected.…”