Influence of initial phase on subharmonic resonance in an incompressible boundary layer

The influence of the initial phase fundamental and subharmonic waves on resonance is investigated for an incompressible boundary layer with zero adverse pressure gradients. Parabolized stability equation analyses are carried out various combinations phases waves. amplification higher modes found to depend strongly phases, this dependence consistent observations from previous experimental studies. There exists a certain combination that leads or anti-resonance condition (i.e., maximum minimum growth, respectively). For all appears be function single parameter represents difference between in resonant interaction depends rather than individual wave. In downstream direction, changes value eventually converges specific approximately ranging 80° 90°, regardless difference. This transient behavior does not start until wave enters parametric stage, which yields double-exponential growth. qualitative characteristic remains unchanged frequencies spanwise wavenumbers as well gradients studied. method analysis results contribute physical foundations controlling boundary-layer transition dominated by resonance.