三维曲面边界层的简化线性稳定性理论

Simplified linear stability theory for boundary layer of three-dimensional curved surface

分析了贴体曲线正交坐标系下线性化小扰动方程(LPEs)中各曲率项的量级,结果显示大部分曲率项都是高阶小量。通过忽略这些高阶小量,得到了贴体曲线正交坐标系下的简化版小扰动方程。简化版小扰动方程不需要繁琐推导,可直接来源于笛卡儿坐标系下的小扰动方程的简单拓展,方便初学者使用。基于简化版的小扰动方程,发展出了能适用于三维曲面边界层的简洁形式的线性稳定性理论(LST)。针对亚声速后掠翼边界层的稳定性,简洁形式LST预测的横流失稳增长率与完整形式LST结果的绝对偏差小于4×10-5;针对高超声速尖锥边界层的稳定性,简洁形式LST精确预测了周向曲率对Mack模态中性曲线的影响。数值结果表明简洁形式的LST拥有与完整形式的LST几乎相同的精确度。

The magnitude orders of the terms of the linearized perturbation equations(LPEs)in bodyfitted three-dimensional curvilinear orthogonal coordinates were analyzed.Results showed that most of the curvature-related terms were represented by high-order small quantities.The LPEs in body-fitted curvilinear orthogonal coordinates were greatly simplified by neglecting the high-order small quantities,which resulted in the simplified LPEs.The tedious derivation of the fully LPEs in curvilinear orthogonal coordinates was no longer needed.The simplified LPEs can be easily obtained by extending the Cartesian LPEs to body-fitted curvilinear orthogonal coordinates for the convenience of beginners.The simplified linear stability theory(LST)suitable for boundary layers with surface curvatures was developed based on the simplified LPEs.The accuracy of the simplified LST was validated in the boundary layers on the curved surfaces of a subsonic sweep wing and a hypersonic sharp-nose cone.For the former,the growth rate of crossflow instability predicted by the simplified LST deviated from the result predicted by the full form LST less than 4×10−5;for the later,the simplified LST accurately predicted the effect of circumferential curvature on the neutral line of the Mack mode.Numerical results showed that the simplified LST had almost the same accuracy with the full form LST.