Aeroacoustic response of a slit-shaped diaphragm in a pipe at low Helmholtz number, 1: quasi-steady results

The aeroacoustic response of a diaphragm in a pipe is studied by means of an analytical model and experimental measurements. The study is restricted to quasi-two-dimensional diaphragms with a sharp-edged rectangular aperture at conditions for which the acoustic source region can be considered compact. The compactness of the source can be realized under two conditions: either a low Strouhal number and a jet Mach number of the order unity; or a low jet Mach number and a Strouhal number of order unity. In this paper, the focus is on the first case. The second case of low Mach number and Strouhal number of order unity is discussed in a companion paper. The results of a quasi-steady theory are compared with measurements of the reflection and transmission coefficients of a diaphragm. The theoretical model is based on Ronneberger's model for a step-wise expansion (D. Ronneberger 1967, Acustica19, 222–235) and Bechert's description of an orifice used as an anechoic pipe termination (Bechert 1980 Journal of Sound and Vibration70, 389–405). An important phenomenon associated with the flow through a diaphragm is the so-called vena contracta effect. This effect is analyzed theoretically as a function of diaphragm opening and jet Mach number by using analytical results for a Borda tube. This allows the use of the theory up to Mach numbers of unity in the free jet downstream of the diaphragm. It is shown that at low frequencies the model and the experimental results are in good agreement. Significant deviations appear only when the Strouhal number reaches unity.