The linearized treatment of general forced gas oscillations in tubes

Abstract

A general linear theory is presented to describe oscillatory flows of gases and liquids in a tube of circular cross-section, including the effects of radial and tangential pressure gradients as well as the temperature. The basic equations are solved by separation of variables. The resulting eigenvalue equation is extensively discussed whereby the splitting of the eigenvalues into ‘bands’ is obtained in a natural way. A systematic analysis of a number of simplified cases leads to analytic approximations for the eigenvalues over an extended domain of parameter variation (frequency, friction) so that a complete survey of all the eigenvalues is established. Then the problem of satisfying simultaneously arbitrary end-conditions for all flow variables with the obtained bands of eigenfunctions is formulated in a way to allow the application of Galerkin's method. Finally the theory is applied to a few examples of ‘end-layers’ and radial resonance, which cannot be treated by previous theories.