An Extension Of The PSE Approach And Its Applications To Nonlinear Evolution Of Multiple-Frequency Perturbations In Shear Layers

An extension of the PSE approach and its applications to nonlinear evolution of multiple-frequency perturbations in shear layers. Focus on the PSE approach to study the nonlinear evolution of three-dimensional disturbances in incompressible boundary layers on flat-plate. Firstly, we used the PSE method to study the subharmonic resonance in the experiment, and compared PSE results with the experimental data. During the comparison process, we found some new problems. The spanwise number which meets subharmonic resonance is 0.352 in the experiment of Kachanov and Levchenko, but as to our theoretical study, the spanwise number is 0.193. The gap between the two spanwise numbers is too large to be acceptable. So we used PSE method to study the growth of three-dimensional disturbance waves in the resonant-triad with different spanwise numbers. Subharmonic resonance is a special case in the rapid growth of the three-dimensional disturbance waves, so we wanted to establish a more general theory, which called "interaction of phase-locked modes" According to our studies, the specific conclusions are as follows:1. The numerical results get by PSE match the experiment result basically in quantitatively.In the qualitative comparison, the plane wave experiences a linear exponential growth in the beginning stage and parametric-resonance stage, and the subharmonic oblique waves also experience a linear exponential growth in the initial stage. But when the nonlinear effects is relatively strong, the subharmonic oblique waves experience a super exponential growth that is faster than the exponential growth predicted by the linear theory, and give a reaction to the plane wave to make it increase again. This result is a consistent with the theoretical description.2. We change the spanwise numbers to study the growth of the three-dimensional disturbance waves in the resonant-triad. We find that the case of ?= 0.352 which meets resonant-triad in the experiment whose the real critical-layer yc,eff (the normal position which the basic flow equal to the corrective phase-velocity)is almost the same with the ymax (the normal position which the maximum of the disturbance) at the downstream position. But as to the the case of ?= 0.193 which meets resonant-triad in the theory whose the real critical-layer yc,eff (the normal position which the basic flow equal to the corrective phase-velocity) has a big difference with the ymax (the normal position which the maximum of the disturbance) at the downstream position. In addition, we alse find that although the phase velocity between a planar and a oblique T-S wave is not the same in the initial location, after adjustment, the real phase-velocity of the three-dimensional disturbance will close to the phase-velocity of the two-dimensional disturbance ultimately.3. As to the mechanism of "interaction of phase-locked modes", if the phase velocity between a planar and a oblique T-S wave is the same phase, then the three-dimensional disturbance waves will experience a super-exponential growth in the theory. But with the PSE results, we find that there is no need the same phase velocity between a planar and a oblique T-S wave in the "interaction of phase-locked modes". we observe the three-dimensional disturbance wave whose phase velocity had a little difference with the plane wave alse experience a superexponential growth that is faster than the three-dimensional wave whose phase velocity was the same with the plane wave.