Analysis of a Self-Sustained Thermally Driven Nonlinear Vibration

A pipe closed at both ends contains a pressure-sensitive plane heat source at its midsection. The heat release rate is assumed to be proportional to the pressure fluctuation at the heater. When the feedback coefficient ε is positive, the system is unstable and a self-excited oscillation grows exponentially in time until shock waves are formed in the system. The ensuing self-sustained nonlinear periodic vibration is calculated for the case ε « 1. It is shown that the pressure-time curve at a fixed point in the pipe has the form of a sawtooth function, while the velocity-time curve has the form of a square wavefunction. To the order of ε, the frequency of the self-sustained vibration is equal to the resonance frequency of the system. The shock waves propagate with constant strength and the shock loci in the xt space are calculated. The effects produced by a time lag in heat source's response are discussed.