Nonlinear distortion of travelling waves in variable-area ducts with entropy gradients

This paper presents the effects of variable-area ducts and axial temperature gradients on the nonlinear distortion of a travelling wave. Quasi-one-dimensional continuity, momentum and energy equations for isentropic flow are solved using a wave front expansion technique. Anevolution equation for the slope of the wave front is obtained. This is a nonlinear ordinary differential equation that can be integrated to obtain a solution in closed form for the slope of the wave front. The solution may admit a singularity for compression wave fronts. The analysis considers pure compression and pure expansion travelling waves. A general criterion is developed for the steepening of a compression wave front into a shock. A general formula is obtained for the location of shock formation. The effects of area variation and axial temperature gradient, and their combined effect on the nonlinear distortion of travelling waves are studied. A number of examples highlighting these effects are presented in the paper.