The laminar–turbulent transition of a supersonic boundary layer induced by an isolated roughness element is investigated using direct numerical simulation, BiGlobal linear stability theory, and the three-dimensional parabolized stability equation. Cylindrical and diamond-shaped roughness elements are investigated in combination with different wall temperatures. Direct numerical simulations show that the cylinder configuration induces an earlier transition than the diamond configuration, with the interaction between the separated shear layer and the counter-rotating vortices causing the transition. BiGlobal analysis and the parabolized stability equation confirm the existence of two unstable instability modes in the wake region, namely, a symmetric mode and an antisymmetric mode, with the former being strongly dominant. The wall cooling and heating effects are studied by changing the wall temperature. Wall heating lifts the inlet boundary layer and weakens the separated shear layer. This, in turn, weakens the wake instability and delays the transition. The antisymmetric model disappears as the wall heating increases. Wall cooling accelerates the transition by enhancing the distortion of the roughness wake, resulting in stronger instabilities. Finally, the Reynolds number based on the momentum defect is used to define the transition criterion, and this is found to be in good agreement with the present simulation results.

使用直接数值模拟、BiGlobal 线性稳定性理论和三维抛物化稳定性方程研究了由孤立粗糙度单元引起的超音速边界层的层流-湍流转变。结合不同的壁温对圆柱形和菱形粗糙度元素进行了研究。直接数值模拟表明，圆柱体构型比菱形构型引起更早的转变，分离的剪切层和反向旋转涡流之间的相互作用引起转变。 BiGlobal分析和抛物化稳定性方程证实了尾流区存在两种不稳定的不稳定模式，即对称模式和反对称模式，其中前者占主导地位。通过改变壁面温度来研究壁面的冷却和加热效果。壁加热提升了入口边界层并削弱了分离的剪切层。这反过来又削弱了尾流的不稳定性并延迟了转变。随着壁加热的增加，反对称模型消失。壁冷却通过增强粗糙度尾流的变形来加速转变，从而导致更强的不稳定性。最后，使用基于动量缺陷的雷诺数来定义转变准则，这与当前的模拟结果非常吻合。