Flow through the reed channel of a single-reed wind instrument
Avraham Hirschberg, Adrianus J. M. Houtsma, A. P. J. Wijnands, H. J. Dane, Joël Gilbert · The Journal of the Acoustical Society of America · 1990
An analytical model of the stationary flow in the aperture of a single reed woodwind instrument is presented [A. Hirschberg et al., Acustica 70, 146-154 (1990)]. For large reed apertures, flow separation occurs on the entire channel length and the volume flow can be calculated by applying Bernoulli's equation with a yens contracta factor of 0.5. For ratios of reed channel length/height of four or more, reattachment occurs and the volume flow is predicted within 5% by using a modification of the theory of Ishizaka and Matsudaira [“Fluid mechanics consideration of vocal chord vibration,” SCRL Monograph 8, Speech Comm. Res. Lab., Inc., Santa Barbara, CA (1972)]. The force on the reed due to the low pressure in the separation region increases with the channel height because the length of the separation region increases with the height. Hysteresis (due to nonlinearity or unsteadiness) in the transition from a fully separated flow to a reattached flow is responsible for oscillations observed in the absence of acoustical feedback. In a similar way, hysteresis in the Bernoulli force may explain oscillations of lips or vocal chords.