Self-Sustained Localized Structures in a Boundary-Layer Flow

When a boundary layer starts to develop spatially over a flat plate, only disturbances of sufficiently large amplitude survive and trigger turbulence subcritically. Direct numerical simulation of the Blasius boundary-layer flow is carried out to track the dynamics in the region of phase space separating transitional from relaminarizing trajectories. In this intermediate regime, the corresponding disturbance is fully localized and spreads slowly in space. This structure is dominated by a robust pair of low-speed streaks, whose convective instabilities spawn hairpin vortices evolving downstream into transient disturbances. A quasicyclic mechanism for the generation of offspring is unfolded using dynamical rescaling with the local boundary-layer thickness.

Figure 1

Snapshot of the localized structure at $t=1050$. Isosurfaces of streamwise velocity perturbation $u=-0.08$ [dark gray (blue)] and ${\lambda }_2=-{10}^{-5}$ [light gray (green)]. The robust core of the structure consists of streaks and streamwise vortices, indicated as well as the early stage of a streak instability. Out of the streak instability grow secondary structures that move downstream and dissipate. The spanwise lines correspond to $x=600$, 800, and 1000. The developing boundary layer is sketched out-of-scale on the left of the figure.

Figure 2

Top: Rescaled vorticity ${\Omega }_x$ vs $t$ and $({\delta }_{0}t/{\delta }^{*})$. Bottom: Projection on the $({\Omega }_x,{\Omega }_y)$ space. The edge trajectory is plotted as solid red lines, and the diverging trajectories are plotted in dotted blue lines. The symbols corresponds to times $t=550$ (cross), 651 (open square), 1050 (filled square), 1350 (open circle), and 1640 (filled circle). The cyclical phase $650\le t\le 1640$ is plotted in thicker red lines.

Figure 3

Side and top view of the quasicycle in rescaled spatial units. Four snapshots are shown, corresponding to the nonrescaled times $t=650$, 1050, 1350, and 1640 marked in Fig. 2 (from left to right), with the same color code as in Fig. 1. The $x$ direction is squeezed by a factor of 2 for better visibility, with the flow from left to right.

Figure 4

Convection velocity $U_c/U_{\infty }$ of perturbations vs $y/{\delta }^{*}$ (shaded area). The Blasius profile is shown (dashed line). The dotted line represents the position at which the Blasius profile represents $99\%$ of $U_{\infty }$.