Receptivity To Slow Acoustic Waves Over Supersonic Boundary Layers Using Parabolized Stability Equations

In this paper, we studied the mechanisms of the receptivity of the supersonicflat-plate boundary layer to slow acoustic waves by using parabolized stabilityequations (PSE). Firstly, based on the characteristics of the slow acoustic wave bothin viscous and inviscid flow field as well as the dispersion relation, The calculationgrid and boundary conditions are adapted. The evolution of a single slow acousticwave was calculated using linear parabolized stability equations (LPSE) in thenon-uniform flow field, whereas the modified non-uniform grid and uniform grid areemployed. The results of two different grids are consistent respectively. Projectionmethod show a single slow acoustic wave cannot induce T-S wave.Otherwise,according to the receptivity theory, the evolution of T-S waves in a flat-plateboundary layer using nonlinear parabolized stability equations, the results are listedbelow:1. The grid and upper boundary conditions need to be amended whencalculating slow acoustic wave. The grid in normal direction should be fromdense to sparse within boundary layer, and uniform outside the boundarylayer. The upper boundary conditions should include the incident andreflected.2. The downstream evolution of the slow acoustic wave and T-S wave withsame frequency were calculated by linear parabolized stability equations.When the two waves have the closest phase velocity, the projection methodis used to analysis the slow acoustic wave. It is indicated that there is seldomT-S wave in slow acoustic wave. It showed that slow acoustic wave is unableto induce the T-S wave.3. The initial disturbance was chosen under the receptivity theory, the evolutionof disturbances were calculated using nonlinear parabolized stabilityequations. The growth of disturbance amplitude varies with frequency andspanwise wavelength. The higher frequency and smaller spanwisewavelength wave are in favor of stimulating high-frequency instability waves.It is found that there are only a small part second mode waves in thehigh-frequency disturbances waves. The result demonstrates that the high-frequency disturbances waves do not belong to the second modeinstability waves.