Study On The Method Determining The Lower Branch Of The Second Mode Neutral Curve Based On APSE

The transition prediction of the hypersonic boundary layer has always been an important issue for the aerodynamic design of high-speed aircraft.In hypersonic boundary layer flow,due to the non-parallel effect and intermodal exchange in the second mode excitation process,so when conducting transition prediction research,and determining the lower branch of neutral curve,we need to consider both of these impacts.In this paper,we take the supersonic flat boundary layer as the research object.First,we discussed the definition of boundary layer stability under parallel and nonparallel flow conditions,and clarified the influence of different normal and growth rate conventions on stability calculation results under the PSE framework.That provides a theoretical basis for APSE to accurately calculate the disturbance.Then,we use the linear parabolic stability equation and the linear Navier-Stokes methods to solve the lower branch of the neutral curve which considering the receptivity.There are two major difficulties in the solution of LPSE,so we came up with a method which based on the adjoint parabolic stability equation in this paper,and use it to determine the lower branch of the neutral curve and consider the receptivity.And we use this method to research the evolution of the second mode disturbance in the boundary layer of the hypersonic plate,and compare its calculation results with the distribution of the growth rate given by the LPSE,which proves that the method based on APSE is reliable.Finally,this method is used to research the process of exciting the second mode under different operating conditions.And we compare the calculated results with the prediction results which given by the linear stability theory and LPSE.Through the above research,we have reached the following conclusions:1.Under the PSE framework,the different normalization and different definitions of growth rate actually only affect the specific value of the rounded part,do not change the slow-changing conditions of the shape function,and do not fundamentally change the calculation results.2.In this paper,we selected three typical case in which fast and slow modes to simultaneously active the second mode,and use the method based on APSE to calculate the position of the lower branch of the neutral curve.And the results were compared with the predictions which from the LST and LPSE methods.From the comparison results,we can see that the position of the lower branch of the neutral curve given by this method is in good agreement with the result given by the LPSE method which given the dangerous discrete mode as the entrance disturbance.This proves that our method can more easily give the position of the lower branch of the neutral curve of the second mode and consider the receptivity.At the same time,it does not depend on the selection of the initial value of the entrance and the calculation efficiency is higher.3.Using the method based on APSE to study the influence of the total temperature and wall temperature on the evolution process of Mack mode excited by the fast and slow mode and the position of the lower branch of the neutral curve of the second mode.From the calculation and analysis results,when considering the receptivity and non-parallelism,the lower branch of the neutral curve given by APSE is generally closer to the upstream than given by LST,indicating that the transition instability process occurs earlier.