Reliability of the eN Method Applied to Hypersonic Blunt Cone Boundary Layers for Transition Prediction
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摘要: eN方法基于扰动在边界层中线性演化过程中的幅值增长程度来预测转捩。以来流Mach数为6、不同壁面温度条件下不同钝度圆锥为研究对象，结合直接数值模拟和抛物化稳定性方程，从eN方法是否能够准确描述扰动在上述边界层中线性增长的角度，分析了该方法预测转捩的可靠性。研究结果表明，在小钝度或高壁面温度情况下，扰动在向下游的演化过程中从第1模态转变为第2模态，基于线性稳定性理论的eN方法变得不再可靠。壁面温度相同，头部钝度越大，eN方法越可靠；同等钝度下，壁面温度越低，eN方法越可靠。由于存在模态转换时，线性稳定性理论总是低估扰动的增长，因而对于给定的转捩判据NT（可由某一工况实验标定给出），若钝度减小或壁面温度增加到一定程度，eN方法给出的转捩位置比实际情况更靠后。重新标定转捩判据时，钝度越小，壁面温度越高，NT的修正程度就越大。Abstract: The eN method predicts transition based on the level of the linear amplitude amplification of the disturbances in the boundary layer. Boundary layers over cones at Mach 6 were investigated with different nose bluntness and under different wall temperature conditions. Combined with direct numerical simulation (DNS) and parabolized stability equations (PSE), from the viewpoint whether the eN method is able to accurately describe the amplification of the disturbances in the above boundary layers, its reliability for transition prediction was interrogated. Results show that in the cases of small bluntness or high wall temperature, the disturbances undergo the intermodal exchange from the first mode to the second mode when travelling downstream in the boundary layer, so that the eN method based on linear stability theory becomes less reliable. Under the same wall temperature condition, as the nose bluntness increases, the eN method is more reliable. For the same nose bluntness, as the wall temperature decreases, the eN method is more reliable. Since linear stability theory always underestimates the amplification of the disturbances when there is an intermodal exchange, for the given transition criterion NT, which could be calibrated by a certain case, as the bluntness decreases or the wall temperature increases to some extent, the eN method tends to produce a further downstream transition location than reality. To recalibrate the transition criterion, the smaller the nose bluntness or the higher the wall temperature, the larger the modification of NT factor.
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Key words:
- hypersonic /
- boundary layer /
- linear stability theory /
- transition prediction /
- eN method
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图 4 绝热壁下不同钝度圆锥的流动剖面
Figure 4. Profiles for adiabatic cones with different nose bluntness
图 5 绝热壁和等温壁Tw=300 K工况下流动剖面的比较(Rn=2.54 mm)
Figure 5. Comparison of flow profiles between cones with adiabatic wall and isothermal wall of Tw=300 K (Rn=2.54 mm)
图 7 DNS壁面压力脉动与PSE得到的幅值演化的比较
Figure 7. Wall pressure fluctuations of DNS compared with amplitude amplification obtained by PSE
图 8 PSE和LST计算得到的不同频率不稳定波的N值曲线(粗实线和虚线分别为PSE和LST的N值包络线)
Figure 8. N factor curves of unstable modes with different frequencies obtained by PSE and LST(thick solid and dashed lines represent the envelope of N factor curves for PSE and LST, respectively)
图 10 不同壁面温度条件下的中性曲线对比(Rn=0.79 mm)
Figure 10. Comparison of neutral curves under different wall temperature conditions(Rn=0.79 mm)
图 11 PSE和eN方法的N值曲线对比(Rn=0.79 mm)
Figure 11. Comparison of N factor curves obtained by PSE and eN method(Rn=0.79 mm)
表 1 PSE和eN方法给出的转捩位置的比较
Table 1. Comparison of predicted transition locations obtained by PSE and eN method
Tw/K Rn/mm NT=4.5 NT=10 PSE/m eN/m ΔξT/m PSE/m eN/m ΔξT/m adiabatic 0.025 4 0.09 0.16 0.07 0.28 0.51 0.23 adiabatic 0.79 0.23 0.24 0.01 0.49 0.60 0.11 adiabatic 2.54 0.60 0.60 0.00 0.95 0.95 0.00 300 0.025 4 0.08 0.11 0.03 0.26 0.37 0.11 300 0.79 0.19 0.19 0.00 0.39 0.40 0.01 300 2.54 0.51 0.51 0.00 0.75 0.75 0.00 
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表 2 绝热壁下eN方法用于不同钝度工况所需的N值修正
Table 2. Modification of N factor in eN method for adiabatic cones with different nose bluntness
Rn/mm NT=4.5 NT=10 PSE: ξT/m eN: ΔN PSE: ξT/m eN: ΔN 0.025 4 0.09 2.25 0.28 3.95 0.79 0.23 0.35 0.49 1.25 2.54 0.60 0.00 0.95 0.00
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表 3 不同壁面温度下eN方法所需的N值修正(Rn=0.79 mm)
Table 3. Modification of N factor in eN method for cases with different wall temperatures (Rn=0.79 mm)
Tw/K NT=4.5 NT=10 PSE: ξT/m eN: ΔN PSE: ξT/m eN: ΔN 300 0.19 0.00 0.39 0.25 400 0.22 0.10 0.47 0.80 adiabatic 0.23 0.35 0.49 1.25 500 0.25 0.39 0.50 1.85
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