This paper investigates the numerical properties of the DLR TAU code as a tool for combustion instability research. Starting from simple ideal gas test cases of closed cavities, frequencies and damping properties of acoustic eigenmodes are determined and compared to experimental and analytical results. Different methods for calculating damping properties and eigenmode extraction are shown and discussed. A detailed study of the TAU boundary conditions shows their acoustic properties and equivalent acoustic type. Also, results for a simplified partially-reflecting boundary condition are shown. Realistic three-dimensional transversal modes and their damping properties are investigated using a cold-flow combustion chamberlike geometry. The results are compared to experimental and numerical results from the literature. The study concludes with numerical results of a combustion chamber injector coupling in a breadboard experiment designed as a future benchmark experiment to deliver suitable data for CFD validation.