10.4. Cartesian normal mode stability analysis

Similar to the azimuthal stability analysis, it is possible to expand a \(N\)-dimensional Cartesian problem to an \(N+1\)-dimensional problem for stability analysis. Instead of the azimuthal angle \(\phi\), now a wavenumber \(k\) in the \(N+1\)-th direction is introduced. Again, we have a stationary solution of our unknown vector \(\vec{U}^{(0)}(x_1,\ldots, x_N)\), which is independent of \(x_{N+1}\). We then investigate perturbations like \(\epsilon\vec{U}^{(k)}\exp(ik x_{N+1}+\lambda_k t)\).

Analogous to the azimuthal component in azimuthal stability analysis, all vector fields \(\vec{v}\) will get an additional component, i.e. a contribution \(v_{N+1}\vec{e}_{N+1}\). Main differences are that \(k\) is real-valued (opposed to the integer-valued azimuthal mode \(m\)) and that there are no specific boundary conditions for the eigenfunction at an axis of symmetry. However, global degrees of freedom, e.g. a global pressure enforcing a volume, will be deactivate by default when \(k\neq 0\).