Abstract

This paper examines a marginal condition of instability of thermoacoustic oscillations of a gas in a tube with one end open and the other closed by a flat wall, subjected to a smooth temperature distribution axially. Assuming a boundary layer is thin compared with the tube radius, the linear theory is developed in the framework of the first-order theory in its thickness. An idea of the method of renormalization enables us to obtain analytical solutions and to derive the marginal condition when the temperature distribution is parabolic. Solving the condition numerically, the marginal curve for the temperature ratio is displayed graphically against the tube radius relative to the boundary-layer thickness. It is found that the temperature ratio has a minimum and that the curve has two branches with respect to the minimum. While the left branch for viscous mode extends to infinity, the right branch, close to the curve for neutral oscillations, asymptotes a certain temperature ratio as the tube radius increases. Such results should be compared with the ones obtained by Rott in the case of a step distribution. Spatial mode of oscillations is also displayed in the marginal state and some discussions on the energy balance are included.

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