Finite amplitude longitudinal gas oscillations in a duct with temperature gradient

This work investigates the problem of finite amplitude longitudinal oscillations in a straight duct with an initially present temperature gradient. The one-dimensional gasdynamic equations with appropriate boundary conditions specified at both the ends are solved using Galerkin method. The formulation leads to a system of coupled nonlinear ordinary differential equations in time. These equations are solved numerically from the initial state till the limit cycle. An example of constant temperature gradient is considered. When the temperature is uniform in the duct, the limit cycle pressure waveform has a shock that decays over a sawtooth profile. The presence of non-zero temperature gradient in the duct reduces the shock strength and smoothens the sawtooth profile. At high values of temperature gradient, both the shock and sawtooth profile disappear. The shape of waveform becomes like an inverted U. In the presence of temperature gradient, the frequency response curve is skewed towards low frequency showing the softening behavior. The magnitude of softening increases with the increase in temperature gradient.