82 / 2019-09-30 17:27:08

A study on the disturbance evolution and the transition by resonant-triad interactions with a side-frequency disturbance in a boundary layer

Side-frequency disturbance,dominant-frequency disturbance,resonant-triad,parabolized stability equations (PSE),“Steady-Spanwise-Wave-Working (SSWW)” mechanism

In usual studies on disturbance evolutions and resonant-triad interactions in the boundary layers, the dominant-frequency disturbances are pure, implying that the side-frequency disturbances are neglected. However, in practical engineering problems, there are always side-frequency components, whose frequencies are close to those of the dominant-frequency waves. If the amplitudes of the side-frequency disturbances are high enough, they could affect the evolutions of the dominant-frequency ones, and the detailed process can be described as follows. The nonlinear interactions between the side-frequency and dominant-frequency disturbances generate difference-frequency disturbances. The difference-frequency perturbations are supposed to be quite “dangerous” because of their low frequencies. The difference-frequency waves and their harmonic waves can interact with each wave in the flow field such that the frequency bandwidths of all waves are broadened. Consequently, the evolutions of the dominant-frequency waves are modified. For the transition caused by resonant-triad waves, the side-frequency perturbations could change the transition location.
In this paper, the parabolized stability equations (PSE) are employed. The method is verified by comparison with the direct numerical simulations (DNS). Then the method is used to study the influence of a side-frequency component on the development of a dominant-frequency disturbance. The numerical results are qualitatively consistent with the experimental data of Kachanov et al. and Saric & Reynolds, and are consistent with the theoretical results of asymptotic analysis of Nayfeh & Bozatli. In each comparison case, the side-frequency wave is weaker than the dominant-frequency one. However, when its amplitude becomes high enough, it induces strong a difference-frequency wave, and influences development of the dominant-frequency wave.
In order to study the influence of the side-frequency perturbation on the transition caused by resonant-triad interactions, one has to calculate a transition case without side-frequency wave as a reference. Herein, it is found that the resonant-triad waves and the mean flow distortion cannot trigger transition by themselves. A “Steady-Spanwise-Wave-Working (SSWW)” mechanism is necessary to cause transition, in that the steady spanwise waves generated by the nonlinear interaction between the pair of three-dimensional (3D) waves play an indispensable role in transition. Then we investigate the transition caused by resonant-triad interactions with a two-dimensional (2D) side-frequency component of the 2D dominant-frequency wave. The side-frequency wave makes transition earlier, and the stronger the side-frequency component is, the earlier the transition takes place. The relative amplitude of the side-frequency disturbance, i.e. the ratio of its absolute amplitude to that of the dominant wave, plays the essential role in the transition advance, while its absolute amplitude has no essential effect. Furthermore, if the relative amplitude reaches the threshold level of 40%, the transition location can be affected substantially. For the transition with the side-frequency component, the “SSWW” mechanism still works, and the side-frequency perturbation enhances the effects of the “SSWW” mechanism, such that the transition occurs earlier.