Transient growth on the homogenous mixing layer

We compute the three-dimensional (3D) optimal perturbations of an homogeneous mixing layer. We consider as a base state both the hyperbolic tangent (tanh) velocity profile and the developing two-dimensional (2D) Kelvin-Helmholtz (KH) billow. For short enough times, the most amplified perturbations on the tanh profile are 3D and result from a combination between the lift-up and Orr mechanisms[1]. For developing KH billows, there are different mechanisms that prevail depending on the initial amplitude of the billow, the spanwise wavenumber and the time of the response observed. We determine when the largest transient growth at a particular time is associated with an optimal response reminiscent of the elliptic or hyperbolic instability.