Numerical Study of Thermoacoustic Spontaneous Oscillations in an Axisymmetric Closed Tube

We study the stability of thermoacoustic spontaneous oscillations (Taconis oscillations) by numerically solving the axisymmetric compressible Navier-Stokes equations. The flow fields in a cylindrical closed tube are simulated by the block pentadiagonal scheme with the second--order time marching and the fourth-order convective term. We consider the helium gas in the closed tube with both hot end parts and a cold central part. When the temperature ratio is larger than the critical value, the spontaneous oscillation is observed. In the case of $r$(the length ratio of the hot part to the cold part)=1, we observe the fundamental mode of a standing wave as the spontaneous oscillation. On the other hand, in the case of $r < $ 0.42 three different oscillation modes are observed: the fundamental mode and the second mode of a standing wave, and the oscillation with a shock wave.