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.
Highlights
The boundary-layer transition is directly related to the aerothermodynamic performance of flight vehicles, as it causes considerable changes in skin-friction and heat-transfer characteristics.[1]
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
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
Summary
The boundary-layer transition is directly related to the aerothermodynamic performance of flight vehicles, as it causes considerable changes in skin-friction and heat-transfer characteristics.[1]. A typical process of the natural transition consists of receptivity, linear amplification of instability waves, nonlinear evolution of instabilities or secondary instability, and final breakdown.[2,3,4] The transition can be classified into several types depending on the physical mechanism that dominates the nonlinear stage and final breakdown process. Studying the nonlinear stage can provide useful information for the physical understanding of transition phenomena. As described by Borodulin et al.,[5] some resonant interactions of instability modes usually dominate the initial nonlinear stage.[2,6] In early research, the importance of resonant wave interactions was recognized in theoretical studies, including those of the theory of secondary instability.[6,7,8] Craik[9] developed the theory of resonant interaction of three waves in boundary-layer flows on the basis of a weakly nonlinear approach
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