Two-dimensional disturbance growth of linearly stable viscous shear flows

Instability of fluid flow has long been linked with the eigenvalues of the Orr-Sommerfeld equation. Recently, it has been shown that even though all eigenvalues may be stable, it is still possible to have disturbance growth. This is because the Orr-Sommerfeld operator is non-normal. We identify which eigenmodes are important in two-dimensional disturbance growth. We find a relationship between maximum growth and growth caused by the adjoint of the leading eigenmode for both plane Poiseuille flow and plane Couette flow. The work points to a connection between the occurrence of the first degeneracy of the Orr-Sommerfeld operator and the first appearance of disturbance growth.