Abstract
A discussion of sound propagation in a moving fluid confined by cylindrical walls is presented. Based on the continuity equation and the Euler equation, a single “exact” ordinary differential equation in the acoustic pressure is derived for the case where the medium flow v0(r) depends on the radial co-ordinate only and points in the axial direction. This “exact” pressure wave equation is solved semi-analytically by means of the Frobenius method and compared with the conventional approximative wave equation known as the local-plane-wave (LPW) approximation for a range of flow values. In this way, information about mode phase-speed changes with flow and flow-meter performance is obtained. It is found that the LPW approximation works well only for mode propagation parallel or nearly parallel to the direction of flow. Based on the “exact” acoustic pressure wave equation, it is also concluded that flow-meter errors become independent of ultrasound frequency and cylinder radius, a point that the LPW approximation fails to predict. Furthermore, an “exact” procedure shows that flow-meter errors depend on the Reynolds number and the mode number only. In actual fact, it is found that flow measurement based on the fundamental mode is approximately free of errors while all other modes are characterized by the same (and, generally, non-vanishing) deviation of measurement.
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