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

Abstract

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