Saddle-node bifurcations of traveling waves in the asymptotic suction boundary layer flow

This work aims at investigating three-dimensional finite-amplitude traveling wave solutions of the asymptotic suction boundary layer, as well as their role in state space, based on direct numerical simulations. Using a body forcing and allowing wall-normal perturbations at the wall to be nonzero and linked to the pressure gradient through a wall permeability a, nonlinear invariant solutions of the Navier-Stokes equations are found and continued towards low values of the Reynolds number. The solutions, having a reflect symmetry in the spanwise coordinate, set a new threshold in state space for the onset of exact coherent flow structures. The obtained traveling waves emerge from a saddle-node bifurcation at Re-SN = 225 (based on the displacement thickness) for vanishing wall permeability. Increasing a to small but finite values changes but slightly the value of Re-SN. In all cases, the obtained Re-SN is below the lowest Reynolds number at which sustained turbulence is observed, namely, Re approximate to 270 according to recent works. The corresponding waves in the Blasius boundary layer flow without suction exist from Re-SN = 496, meaning that the asymptotic suction boundary layer might be more nonlinearly unstable (although more linearly stable) than the Blasius one. In the interval of Re studied it is found that the traveling waves occupy a region extending from the wall to the log-law region, with maximum root-mean-square velocities approaching those obtained by direct numerical simulations in the turbulent regime, especially for nonvanishing permeability. These solutions are found to be unstable, with steady leading modes, whose growth rate increases with the permeability and with the Reynolds number. When the trajectory escapes from these traveling waves along their unstable directions, transition to turbulence is observed even at Reynolds number as low as Re = 240, suggesting that these solutions may play an important role in turbulent transition at low Reynolds numbers.