The Nonlinear Evolution Of Disturbances In Supersonic Flat-plate Boundary Layers

Characteristics of the nonlinear evolution of first and second modes in flat-platesupersonic boundary layers at Mach4.5、6and8are investigated by using numericalsimulation. Further research on the nonlinear evolution of second modes shows aselective mechanism of Klebanoff-type3-D disturbances with high spanwisewavenumber by the second mode disturbance. The present paper analyzes the linearand nonlinear amplification rate of Klebanoff-type3-D disturbances. The resultsreveal the relationship between the selective mechanism and the amplification rateand spanwise wavenumber of Klebanoff-type3-D disturbances. The results indicatethe range of the spanwise wavenumber of the selected Klebanoff-type3-Ddisturbances. The specific conclusions are as follows:1. The nonlinear evolution of first mode differs from that of second mode. Theamplified disturbances spread from low spanwise wavenumber to high spanwisewavenumber and Λ-vortex arises. It is very similar to the evolution in anincompressible boundary layer.For nonlinear evolution of second mode disturbances, the rapid growth ofordered spanwise small-scale3-D disturbances appear downstream. There ishighly selective enhancement of3-D disturbances with high spanwisewavenumber. This phenomenon is the main characteristic of the nonlinearevolution of second modes in a supersonic boundary layer.2. The selectivity of3-D disturbances by2-D second mode disturbance shows thatnonlinear interaction of Klebanoff-type is stronger than other type, espically forsubharmonic resonance.3. Further research of Klebanoff-type nonlinear interaction shows that the3-Ddisturbances are prompt to amplify when the amplitude of2-D disturbancereaches a certain level. The amplitude of the disturbances which spanwisewavenumber concentrating in a span amplifies highly. This span nearly lies in thejunction of the upper branch and inferior branch of the neutral curve and belongsto the scope of high spanwise wavenumber.4. The relationships between2-D disturbance and Klebanoff-type3-D disturbancesare summarized as follows: When the amplitude of2-D disturbance is low,3-D disturbances evolveaccording to linear stability theory. The higher the spanwise wavenumber of3-Ddisturbance is, the smaller the linear amplification rate becomes. When thespanwise wavenumber is larger than the scope of the neutral curve, the linearamplification rate becomes negative and the disturbance becomes damped wave;When the amplitude of2-D disturbance reaches a certain level, nonlineareffects and the linear amplification rate changed to the nonlinear amplificationrate. Nonlinear amplification rate is related to the amplitude of2-D disturbance.The greater the amplitude of2-D disturbance is, the higher the nonlinearamplification rate becomes. The nonlinear amplification rate is also associatedwith the spanwise wavenumbers of3-D disturbances. The higher the spanwisewavenumber of3-D disturbance is, the larger the nonlinear amplification ratebecomes;The location from the linear amplification rate to nonlinear amplificationrate is not only affected by the amplitude of2-D disturbance, but also influencedby the amplitude of other3-D disturbances. The higher the amplitude of other3-D disturbances is, the sooner the transformation location appears.5. The linear and nonlinear amplification rate of Klebanoff-type3-D disturbancesare related to the spanwise wavenumber, which leads the selectivity of3-Ddisturbance by2-D second mode disturbance. The linear amplification rate of thesmall spanwise wavenumber is high, but the nonlinear amplification rate is low;the linear amplification rate of the high spanwise wavenumber is low, but thenonlinear amplification rate is high. So the overall effect causes the3-Ddisturbances centered at the junction of the upper branch and inferior branch ofthe neutral curve amplified highly.