Generation Of Streamwise Vortex In A Supersonic Flat-plate Boundary Layer

As a basic structure in the process of transition from laminar flow to turbulent flow in a supersonic flat-plate boundary layer,streamwise vortex is of great significance for understanding the generation and evolution of turbulence.Although a lot of experimental and numerical studies on streamwise vortex have been carried out,and the generation and evolution laws have been revealed to some extent,the generation problem of streamwise vortex has not been completely solved up to now.On the problem of how to generate streamwise vortex,the existing researches generally generate streamwise vortex through the nonlinear action of disturbances.Therefore,this paper raised the question:Can streamwise vortex be obtained by linear superposition of disturbances?To answer this question,direct numerical simulation（DNS）was used to simulate the generation process of streamwise vortex in a supersonic flat-plate boundary layer with Ma=4.5.The generation conditions of streamwise vortex were put forward,according to the rotational tolopogy structure of streamlines in the cross section of vortex.And based on the conditions,the effects of different factors on the generation of streamwise vortex were discussed,including the amplitude（A）,frequency（ω）and spanwise wave number（β）of disturbance,Reynolds number and Mach number of the base flow,considering the linear superposition of a pair of oblique waves（with the same A andωbut oppositeβ）.The main conclusions are as follows:1.Streamwise vortex can be generated by a pair of unstable oblique waves in the first mode,and the main components of the disturbance are the（1,1）wave added at the inlet and the（0,2）wave generated by nonlinear action when streamwise vortex appears.2.The generation conditions of streamwise vortex were proposed in this paper:at least one point exists in the streamwise cross section,which satisfies the velocity component in the section is 0,and the characteristic discriminant invariant of the velocity gradient tensor is less than 0.This condition is verified by DNS results.3.Considering the linear superposition of a pair of oblique waves,since the velocity v₀exists in the base flow,according to the conditions above,the generation of streamwise vortex requires the disturbance amplitude A to be greater than a certain threshold A_e.When the inlet amplitude A₀ is greater than A_e,streamwise vortex will be generated at the inlet;when the inlet amplitude A₀ is less than A_e,no streamwise vortex will be generated at the inlet.However,as A grows along the flow direction,streamwise vortex may be generated downstream.4.Considering the linear superposition of a pair of oblique waves,the effects ofωandβon the generation of streamwise vortex include two aspects:First,ωandβaffect the streamwise and spanwise scales of vortex structure respectively.In order to obtain streamwise vortex,the ratio ofω-βshould be small.Second,ωandβwill affect the eigenfunction of the disturbance thus affecting A_e.Increasingωor decreasingβwill decrease A_e.The influence of Reynolds number and Mach number on the generation of streamwise vortex is mainly reflected in the influence on the base flow velocity v₀.By increasing the Reynolds number or decreasing the Mach number,v₀ can be reduced,thus the amplitude threshold A_e required for the generation of streamwise vortex will be reduced.