Numerical simulations of thermoacoustic oscillations in a looped tube by asymptotic theories for thickness of diffusion layers

Thermoacoustic oscillations in an air-filled, looped tube with a stack inserted are simulated numerically by using asymptotic theories for the ratio of a radius of flow passage to a typical thickness of the thermoviscous diffusion layer. The stack is composed of many pores axially and is sandwiched by hot and cold heat exchangers to impose a temperature gradient on the air in each pore. Weakly nonlinear wave equations based on the boundary-layer theory are used for a section in the outside of the stack. In each pore, the diffusion-wave (advection) equation is employed. Matching conditions at both ends of the stack require the conservations of mass, momentum and energy fluxes. An initial-value problem is solved from a disturbance of a pulsed axial velocity along the loop. When the temperature ratio is below a certain value, the initial disturbance is decayed out. However when the ratio exceeds it, it becomes unstable to grow in amplitude. Between the stable and unstable regimes, there exists a marginal sta...