Linear and nonlinear development of localized disturbances in zero and adverse pressure gradient boundary-layers

Localized disturbances in laminar boundary-layers at Reynolds number ${\mathrm{Re}}_{{\delta }^{*}}\text{=950}$ were studied using direct numerical simulation. Instability mechanisms in both an adverse and zero pressure gradient were investigated by introducing three different three-dimensional disturbances. The first disturbance was centered around a pair of oblique waves in Fourier space, the second around a plane wave, while the third was axisymmetric. For small amplitudes, the first disturbance developed into a wave-packet of oblique waves in adverse pressure gradient and into a streaky structure with a trailing wave-packet in a zero pressure gradient. The second disturbance developed into a wave-packet centered around plane waves in both pressure gradients. The third disturbance developed into a wave-packet of plane waves in adverse pressure gradient and, due to the transient growth mechanism, into a streaky structure in a zero pressure gradient. For finite-amplitude plane wave-packets in a zero pressure gradient, a subharmonic secondary instability was observed which subsequently developed into an elongated $\Lambda$ structure. The secondary instability was less significant in adverse pressure gradient due to the large growth rate of the primary instability. Breakdown was observed as high-frequency oscillations on the spike over the head of the $\Lambda$ vortices. Providing the initial amplitude was sufficiently large, the vortex pair yielded the fastest route to turbulence. The main growth mechanism in this scenario, in addition to the exponential growth in the adverse pressure gradient case, was the nonlinear excitation of transient growth of streaks by interacting oblique modes. Disturbances dominated by streaks needed substantially larger amplitudes before secondary instability or breakdown occurred. In that case, breakdown was shifted from the spike to an instability of the rear part of the streaks.