Optimal energy growth in variable-density mixing layers at high Atwood number

We analyze the influence of the specific features of variable-density Kelvin-Helmholtz (hereinafter quoted VDKH) roll-ups on the development of three-dimensional secondary instabilities. We use a direct-adjoint nonmodal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base-flow. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high-Atwood number (up to At = 0.5 here), temporally evolving mixing-layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. It is found that additional mechanisms of energy growth are onset a little before the saturation time of the primary two-dimensional KH instability, corresponding to a substantial accumulated baroclinic spanwise vorticity in the base-flow. Beyond in time, the extra energy gain due to increasing Atwood numbers, relies both on the higher strain rate found in the vorticity enhanced braid and on contributions from spanwise baroclinic sources. Both effect are responsible for the organisation of the perturbation spanwise vorticity into elongated layers along the braid. They are associated to longitudinal velocity streaks that are responsible for the energy growth. This mechanism is boosting the energy gain over the whole range of spanwise wavenumbers but the short wavelength instabilities benefit more from the increase of the Atwood number than the long wavelength ones. Finally, it is observed that all optimal perturbations eventually triggers an hyperbolic-type instability, even at small spanwise wavenumber, where elliptic modes are favoured in the homogeneous case.