局部散射理论在超声速边界层中的应用

The prediction and control of laminar-turbulent transition in supersonic boundary layers are critically important in engineering applications. Although a huge number of numerical and experimental investigations have been carried out based on these issues, and some of the fundamental mechanisms are not yet clear. This project will establish a generic mathematical framework based on the large Reynolds number asymptotic theory, to describe the impact of a small abrupt change at the wall, such as a roughness element, a blow & suction slot, a gap or a corner, on the transition process. The regimes include the receptivity of the instability waves by the interaction between the abrupt change and the freestream acoustic wave, the scattering of the instability waves induced by the abrupt change and the noise radiation due to the scattering process. The framework is proposed by extending the local scattering theory which was developed by the applicant on describing the dynamic motion of the subsonic shear layers, however, some more interesting mechanisms are included in the supersonic flows, e.g. the generation and the rapid distortion of the second Mack mode instability waves. Moreover, in order to verify the accuracy of the supersonic local scattering theory, direct numerical simulation will also be performed in this project.

对超声速边界层从层流到湍流的转捩的预测与控制有重要的工程应用价值。虽然当前已有大量的数值与实验工作关于该问题展开，但很多原理性的机制尚不清楚。本项目拟基于大雷诺数渐进理论，建立一套通用的数学理论框架，来描述壁面处的小突变（包括粗糙元、吹吸、狭缝、折角等）对转捩的影响。具体包括突变与自由流中声扰动相作用而激发不稳定波的感受性问题，突变对不稳定波的散射问题以及由散射引起的噪声辐射问题。该理论框架的基础是申请人前期基于亚声速剪切层提出的局部散射理论，但是，在超声速问题中，更多有趣的机制被包含进来，如第二模态不稳定波的产生与快速畸变机制。此外，本项目还将通过直接数值模拟，验证可压缩局部散射理论的可靠性与精度。

对超声速边界层从层流到湍流转捩的预测与控制是航天飞行器气动设计中的瓶颈问题。从文献来看，虽然已有大量的数值与实验工作关于该问题展开，但很多原理性的机制未解释不清楚。本项目基于大雷诺数渐近理论，建立一套通用的局部散射理论框架，以定量描述壁面处的小突变（包括粗糙元、吹吸、狭缝、折角等）对转捩的影响，并构建考虑局部突变影响的转捩预测模型。对于自然转捩过程，局部突变对转捩的影响主要通过与自由流中扰动相作用而激发失稳模态的局部感受性和线性模态的局部散射两种机制实现。本项目通过多尺度分析法，对超声速、高超声速边界层中由粗糙元引起的局部感受性与线性模态局部散射机制进行了原理性描述。研究发现，局部感受性的首阶贡献来自近壁Stokes层的弯曲效应；线性模态的局部散射是粘性子层的位移效应与壁面几何变形共同作用的效果。基于上述理论对感受性系数与透射系数的预测结果得到了直接数值模拟与谐波型线性化Navier-Stokes（HLNS）方程的验证。该理论框架首次基于第一性原理，建立了粗糙元对自然转捩影响的理论体系，并发展了基于该局部散射理论框架的转捩预测模型。该模型是当前最理性的考虑局部突变影响的转捩预测方法，预计在5-10年内应用于临近空间飞行器气动设计中。