  A 2D temporal incompressible stability analysis is carried out for lobed
jets. The jet base flow is assumed to be parallel and of a vortex-sheet type.
The eigenfunctions of this simplified stability problem are expanded using the
eigenfunctions of a round jet. The original problem is then formulated as an
innovative matrix eigenvalue problem, which can be solved in a very robust and
efficient manner. The results show that the lobed geometry changes both the
convection velocity and temporal growth rate of the instability waves. However,
different modes are affected differently. In particular, mode 0 is not
sensitive to the geometry changes, while modes of higher-orders can be changed
significantly. The changes become more pronounced as the number of lobes N and
the penetration ratio $\epsilon$ increase. Moreover, the lobed geometry can
cause a previously degenerate eigenvalue ($\lambda_n = \lambda_{-n}$) to become
non-degenerate ($\lambda_n \ne \lambda_{-n}$) and lead to opposite changes to
the stability characteristics of the corresponding symmetric (n) and
antisymmetric (-n) modes. It is also shown that each eigen-mode changes its
shape in response to the lobes of the vortex sheet, and the degeneracy of an
eigenvalue occurs when the vortex sheet has more symmetric planes than the
corresponding mode shape (including both symmetric and antisymmetric planes).
The new approach developed in this paper can be used to study the stability
characteristics of jets of other arbitrary geometries in a robust and efficient
manner.
