Nonorthogonality analysis of acoustics and vorticity modes: Should thermoacoustic energy norm be time-invariant?

Detrimental thermoacoustic instability occurs frequently in many propulsion systems, such as rocket motors, and aeroengines. To predict, quantify and analyze thermoacoustic instability, it is critical to choose a proper energy norm of acoustic disturbances in a thermoacoustic combustor. It has been shown that Chu's energy norm is not associated with spurious transient energy growth, when acoustic and vorticity perturbations are assumed to share the same wave numbers. The present work performs a more generalized investigation. It is shown that vorticity and acoustic modes are non-orthogonal, even if their wave numbers are different. Further study reveals that although the vorticity and acoustic modes are not orthogonal, the cross terms of these two flow disturbances do not lead to spurious energy growth.