The stability of the jet in cross-flow is investigated using a complete set-up including the flow inside the pipe. First, direct simulations were performed to find the critical velocity ratio as a function of the Reynolds number, keeping the boundary-layer displacement thickness fixed. At all Reynolds numbers investigated, there exists a steady regime at low velocity ratios. As the velocity ratio is increased, a bifurcation to a limit cycle composed of hairpin vortices is observed. The critical bulk velocity ratio is found at approximately R = 0.37 for the Reynolds number Re D = 495, above which a global mode of the system becomes unstable. An impulse response analysis was performed and characteristics of the generated wave packets were analysed, which confirmed results of our global mode analysis. In order to study the sensitivity of this flow, we performed transient growth computations and also computed the optimal periodic forcing and its response. Even well below this stability limit, at R = 0.3, large transient growth (10 9 in energy amplification) is possible and the resolvent norm of the linearized Navier-Stokes operator peaks above 2 × 10 6 . This is accompanied with an extreme sensitivity of the spectrum to numerical details, making the computation of a few tens of eigenvalues close to the limit of what can be achieved with double precision arithmetic. We demonstrate that including the meshing of the jet pipe in the simulations does not change qualitatively the dynamics of the flow when compared to the simple Dirichlet boundary condition representing the jet velocity profile. This is in agreement with the recent experimental results of Klotz et al. (J. Fluid Mech., vol. 863, 2019, pp. 386-406) and in contrast to previous studies of Cambonie & Aider (Phys. Fluids, vol. 26, 2014, 084101). Our simulations also show that a small amount of noise at subcritical velocity ratios may trigger the shedding of hairpin vortices.

Laminar-turbulent transition prediction is of practical interest in aircraft design since transition affects important aerodynamic quantities such as drag and heat transfer. Extended laminar flow on aerodynamic surfaces is an effective way of reducing aircraft drag. One of the major challenges for the implementation of laminar-flow surfaces is the potential for any irregularity to move transition upstream. Under low-disturbance environment, boundary-layer transition results from the growth and breakdown of different flow instabilities. In 2-D flows the scenario is dominated by Tollmien-Schlichting (TS) instabilities. Common wing-surface irregularities, such as two-dimensional steps, gaps or waviness can alter the growth characteristics of TS waves and therefore must be taken into account at the design stage.

A realistic aircraft wing is not expected to be an ideally smooth surface: the influence of junctions or wing panels must be taken into account in the laminar wing design. The presence of these spanwise invariant two-dimensional surface irregularities further amplifies the boundary-layer streamwise instabilities existing on a smooth surface, potentially causing an early transition to turbulence. In the present work, the effect of steps, gaps and humps on the development of Tollmien-Schlichting (TS) waves in an incompressible boundary layer is studied using direct numerical simulations (DNS). For a specific height we investigated several shapes of the geometric irregularity. Depending on their shape the surface imperfections give rise to a local separation bubble which interacts with the oncoming TS waves: for the frequency considered, all the surface irregularities have a destabilizing effect, with the rectangular hump case being the most dangerous one.