A model for sound propagation in capillary ducts with mean flow

Abstract

A theoretical formulation is carried out of acoustic wave propagation in a narrow capillary tube with steady gas flow. The transverse variations of the particle velocity, temperature, and viscosity are considered. A fully developed laminar steady flow is assumed and the concept of a complex propagation constant is introduced in the formulation. The final equation form reduces to a Kummer-type differential equation and its solution is obtained in terms of confluent hypergeometric functions. The dispersion equation for the complex propagation constants takes on a recursive form. A simplified form of the analysis permits comparison with previous results dealing with visco-thermal effects and includes the features of Poiseuille-type laminar steady flow for low and medium shear wave numbers. Numerical simulation results show that the effect of steady flow is very significant for the backward traveling waves, and the assumption of a parabolic velocity profile for shear wave numbers less than four should be used carefully when the flow Mach number is greater than about 0.1. The present theory is applicable for shear wave numbers up to 10 or more, with the non-parabolic axial velocity fluctuations included, which encompasses almost all the possible practical situations of capillary duct dimension, temperature, and flow velocity. The theory would be useful as an approximation in solving the acoustic problems of the monolith in catalytic converters for automotive exhaust systems and of the propagation of sound in a porous medium (C) 1996 Academic Press Limited