Compressible Weakly Nonlinear Theory Considering Non-parallelism And Its Application

Crossflow instability is one of important instability mechanisms in three-dimensional boundary layers.The nonlinear saturation is an important feature of crossflow instability.The non-parallelism is also a non-negligible factor in many cases,such as swept wings,blunt bodies at high angle of attack,etc.For which,themethod,which is based on the linear stability theory?LST?that can not consider the nonlinear and the non-parallism is invalid in the transition prediction of the crossflow instability.Meanwhile,the nonlinear parabolic stability equations?NPSE?method which considers the nonlinear and the non-parallelsim is inflexible and inefficient in dealing with crossflow instability problem.In order to predict the evolution of crossflow disturbances precisely and efficiently,proper methods need to be adopted.In this paper,the incompressible weakly nonlinear theory?WNST?is extended to the compressible spatial evolution of disturbance case,and the weakly nonlinear theory considering non-parallelism?NEPSE?is built up.In this paper,we describe the detailed derivation and solving processes of both methods.A subsonic infinite-span swept wing is selected as a prototype,and we use these two methods?WNST and NEPSE?to study the nonlinear evolution of crossflow disturbances in the swept-wing boundary layer.The main works and conclusions are shown as below:1.The laminar basic flow of the swept-wing boundary layer is numerically calculated by using the direct numerical simulation method?DNS?,and the flow characteristics are analyzed.The variation of the boundary layer displacement thickness with the streamwise direction is large,indicating that the non-parallelism is strong at the leading edge of the wing.2.We use DNS method to study the evolution of the stationary vortices and travelling waves,and the nonlinear evolution characteristics are obtained.The crossflow disturbances undergo the amplitude saturation stage and the saturated vortices present‘half-mushroom'shape.The direction of vortex axis of stationary vortices is nearly parallel with the direction of potential flow,whereas the direction of vortex axis of travelling wave deviates from the potential flow direction.3.The weakly nonlinear theory of compressible spatial evolution of disturbances is derived.The evolution of the disturbances in the artificially constructed parallel flow and the swept-wing flow are all studied by using WNST method,and their results are compared with the results of the DNS.The results of WNST agree well with the results of DNS in the case of the parallel flow.However,in the case of the swept-wing flow,because of the remarkable effect of non-parallelism on the crossflow instability,there are considerable differences between the results of WNST and DNS.Through the analysis of the nonlinear correction of growth rate,we find that the main part affecting the fundamental wave is the basic flow modification.4.The general form of nth-order EPSE is derived.We use the 1^st-order EPSE?EPSE1?and 2^nd-order EPSE?EPSE2?to analyze the effect of non-parallelism on the growth rate of crossflow disturbances.The results of EPSE1 and EPSE2 are more precise than the results of LST.In the framework of the EPSE,we build the weakly nonlinear theory considering non-parallelism by following the method of WNST.We use the NEPSE method to calculate the evolution of disturbances in the swept-wing boundary layer.The results of NEPSE are compared with the results of the DNS and WNST.It shows that the results of NEPSE is closer to the results of DNS than the results of WNST.5.We study the influence of wave parameters and initial amplitudes on evolution of disturbances by using the NEPSE method.It is found that the increase of frequency not only can reduce the saturation amplitude of the envelop curve formed by use of the variation of spanwise wavenumber,but also can advance the position of the saturation amplitude of the envelop curve.The increase of initial amplitude can also advance the position of the saturation amplitude,but it is hard to change the magnitude of the saturation amplitude.