The Method Of The Stability Analysis And Receptivity To Acoustics In Supersonic Boundary Layers

In this paper, the compressible boundary layer on the flat plate is considered. A stability analysis method is proposed, which can consider the nonparallel of the boundary layer. The mode decomposition method in spatial model is improved. In hypersonic boundary layer, the synchronization point and the branch point of the fast and slow mode are investigated, and the characteristic of the modal exchange during the fast and slow mode is revealed. There is a l ocalized receptivity mechanism between fast and slow acoustics, which is realized by the nonparallel of the boundary layer and the nonlinear interaction. Main results are as follow:1. The EPSE method is proposed to consider the nonparallel of the boundary layer. The method is based on the Taylor series expansion. If the parabolized stability equations are expanded in streamwise and higher order terms are neglected, the EPSE equations are obtained. The EPSE equations and the homogeneous boundary condition constitute the eigenvalue problem. Near the leading edge, the nonparallel is strong. The amplitudes predicted by the LST do not agree well with the results of the numerical simulation, while the results of the EPSE agree well with the results of the numerical simulation. Far away from the leading edge, the nonparallel can be neglected. The amplitudes predicted by the LST and EPSE agree well with the numerical simulation.2. In spatial model, the mode decomposition is improved, which is not enlarged the order of equations. When the disturbance is decomposed, an operator is used in the inner product. The operator is different for the temporal and spatial model. The temporal model is a linear eigenvalue problem, and the operator is simple. The spatial model is a second-order polynomial eigenvalue problem. With the help of the biorthogonal eigenfunction system, the operator is simplified. The improved mode decomposition method is verified by the numerical simulation.3. The synchronization point and the branch point in the modal exchange are investigated. At the downstream, the phase speeds of the fast and slow mode are approximate or equal. The location where the phase speeds of fast and slow mode are close is call the synchronization point. In some parameters, they are not equal, while in other parameters, they can be equal to each other. The demarcation point of the equality and inequality of the phase speed is called the branch point. Near the synchronization point, the modal exchange will occur. When the fast and slow modes go though the synchronization point, the second mode will be excited. The phenomenon of the modal exchange is investigated by the numerical simulation. The mechanism of themodal exchange can be explained by the propagation of the mode in the nonparallel flow.4. In hypersonic boundary layers, a new receptivity process is revealed, which is that fast and slow acoustic through nonlinear interaction can excite the second mode near the lower-branch of the second mode. They can generate a sum-frequency disturbance though nonlinear interaction, which can excite the second mode. This receptivity process is generated by the nonlinear interaction and the nonparallel of the boundary layer. The results indicate that the receptivity coefficient is sensitive to the wavenumber difference between the sum-frequency and the lower-branch second mode. When the wavenumber difference is zero, the receptivity coefficient is maximum. The receptivity coefficient decreases with the increase of the wavenumber difference. It is also found that the amplitude of the sum-frequency grows linearly in the beginning. The growth rate and the growth distance are related to the wavenumber difference between the sum-frequency and the second mode near the inlet. When the wavenumber difference is zero, the growth rate and growth distance are maximum. The growth rate and growth distance decrease with the increase of the wavenumber difference.