Evolution Of Disturbances In Supersonic Flat Plate With A Small Angle Of Corner Boundary Layer And The Correction For EN Method

In order to predict the transition of boundary layer with small wall corner usingeN method, numerical simulations and hydrodynamic stability theory method arecarried out to investigate the disturbances evolution in hypersonic boundary layer forthe cases where the expansion and compression angles are1o,2o,3owith Mach number4.5by solving the classic O-S equation(LST) and N-S equation(DNS). The amplitudeand growth rate distribution along the flow direction with different frequencies indifferent angles have been revealed. The neutral curve of flat plate boundary layerwith a small corner calculated by hydrodynamic stability theory is consistent with theresults of DNS numerical simulation. Based on these findings, the N value should becorrected in the scenario to predict the transition location with a small wall corner byusing the eN method combined with stability analysis and we provide a correctionfactor of N value in this work.Through the calculation and the following conclusions can be drawn:1. The expansion angle makes the neutral curve downward deflections, the ranges offrequency on the unstable disturbance have become narrowed and the leveldepends on the angle of expansion. For1degree expansion angle, the corrected Nvalue is about0.2. When the angle of expansion greater than2degrees, there isno effective growth area after the angle, so it is not needed for a correction for theN value when use the eN method to predict the evolution of the disturbance wave.2. The compression corner of the boundary layer has a large impact for the secondmodel unstable disturbance wave. The compression angle makes the neutral curveupward deflections and the level depends on the angle of compression. Theneutral curve has a relatively small change for the1degree, and there exists aneffective growth range after the angle for the instability wave which in front ofthe angle, therefore it is needed to correct for the N value. For the1degreecompression angle, the correction N values range from-0.5to0.4; when theangle of compression greater than2degrees, there is no effective growth areaafter the angle, so it is not need to be considered the correction for N value.3. The expansion angle makes a little effect for the lower branch of the neutral curveof the first mode unstable disturbance wave, but it has a large impact on the higher branch of the neutral curve. The expansion angle makes the downwarddeflections of the neutral curve. When the angle is larger, neutral curve can evenbe truncated and then the latest unstable disturbance wave will be aroused on thefarther downstream.4. For the compression angle, the higher branch of the neutral curve will be upwarddeflection, and it has negligible effect on the lower branch and the range offrequency becomes wide. The unstable disturbance wave of before angle exists anoperational growth range when the wave across the compression angle, so it isneeded to consider the correction of N value, and pay particular attention whenuse the eN method predicts the transition.5. There exists a phenomenon of the amplitude modulation when calculated theevolution of the first model disturbance wave, which is due to the interaction ofdifferent wave number of the disturbance waves. Different disturbances wereinspired when the disturbance waves through the corner. Nonetheless, theamplitude variation caused by the modulation is small, and it does not affect thedisturbance of the overall growth or decay trend.