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Phase effect on the modal interaction of flow instabilities is investigatedfor laminar-to-turbulent transition in a flat-plate boundary layer flow.Primary and secondary instabilities are numerically studied with 2DTollmien-Schlichting wave and subharmonic 3D oblique waves at various initialphase differences between these two instability modes. Three numerical methodsare used for a systematic approach for the entire transition process, i.e.,before the onset of transition well into fully turbulent flow. The Floquetanalysis predicts the subharmonic resonance where a subharmonic mode locallyresonates for a given basic flow composed of the steady laminar flow and thefundamental mode. Because the Floquet analysis is limited to the resonatingsubharmonic mode, nonlinear parabolized stability equations (PSE) simulation isconducted with various phase shifts of the subharmonic mode with respect to thegiven fundamental mode. PSE offers insights on the modal interaction affectedby the phase difference up to the weakly nonlinear stage of transition.Large-eddy simulation (LES) is conducted for a complete transition to turbulentboundary layer because PSE becomes prohibitively expensive in the latenonlinear stage of transition. The modulation of the subharmonic resonance withthe initial phase difference leads to a significant delay in the transitionlocation up to $\Delta Re_{x, tr} \simeq 4\times 10^5$ as predicted by thecurrent LES. Effects of the initial phase difference on the spatial evolutionof the modal shape of the subharmonic mode are further investigated. Themechanism of the phase evolution is discussed, based on current numericalresults and relevant literature data.

Effects of Phase Difference between Instability Modes on Boundary Layer Transition