A nonlinear analysis of a simple thermoacoustic system. Ronald E. Kumon (Applied Research Laboratories, University of Texas at Austin, 10000 Burnet Rd., Austin, TX 78713)
A simple thermoacoustic system was studied to try to better understand the interaction between the temperature, pressure, and velocity modes of the system. The system considered was a one-dimensional "tube," closed and isothermal at both ends and filled with a helium gas. Initially, the gas is static but with a sinusoidal temperature distribution. To obtain a simplified model of the system, a Galerkin-type method was applied to the full hydrodynamic equations in one spatial dimension and the ideal gas law equation of state. By substituting highly truncated sine and cosine series in the spatial variable with time-dependent amplitudes into the aforementioned PDEs, the model was reduced to a set of coupled nonlinear ODEs. First, these equations were linearized and examined for series expansions with different number of terms. Next, the nonlinear ODEs were studied. Finally, these results were compared with direct finite-difference calculations using MacCormack's method to integrate the full hydrodynamic equations.