Reynolds and Strouhal numbers in woodwind musical instruments

Abstract

It is convenient to use dimensional analysis in investigating nonlinear effects in woodwind instrument air columns. One dimensionless quantity characterizing the flow is the Reynolds number. There is both a time‐varying particle velocity as well as a time‐independent streaming velocity within the woodwind, and the Reynolds number R defined in terms of the streaming velocity may be written as R ≃ d2/δ2. The time‐varying particle displacement is d and the viscous boundary layer thickness is δ. R ≪ 1 corresponds to the type of streaming first considered by Rayleigh, whereas the streaming at larger values of R is more complicated. The Strouhal number S associated with the oscillating flow and characteristic length L is S = L/d. The criterion that the nonlinear convective acceleration term (v∇v) in the Navier‐Stokes equation be small relative to the linearized acceleration is that S ≫ 1. The particle displacement has been estimated at a fortissimo dynamic level on a flute, clarinet, and alto saxophone by means of pressure level measurements at the first open tone hole and the known radiation impedance of the hole. The streaming Reynolds number is approximately 0.2 for the flute and in the range 9–20 for the clarinet and sax for low register tones at the frequency of the repetition rate. Both R and d rapidly decrease for higher frequency components. The particle displacement ranges from 0.1 mm for the flute to 1 mm for the clarinet and saxophone. The Strouhal number of interest is defined for a characteristic length L on the order of the radius of curvature of the sharp corners within the woodwind bore; for definiteness, I take L = 0.5 mm. The resulting Strouhal number is of order 5 for the flute and 0.5 for the clarinet and saxophone. Since the Strouhal number is not large compared to unity, then the nonlinear effects are important near sharp corners within the bore and particularly so for the clarinet and sax.