Title: Compressibility correction to the k-$ω$ turbulence model that considers the wall-cooling effect Show Chinese title

Title: k-$ω$湍流模型的可压缩性修正，考虑壁面冷却效应

Abstract: In supersonic and hypersonic flows, the near-wall density variation due to wall cooling poses a challenge for accurately predicting the near-wall velocity and temperature profiles using classical eddy viscosity turbulence models. Compressible turbulent boundary layers are known to follow the universal wall law via semi-local transformation. However, developing a turbulence model that predicts a velocity profile, which, via semi-local transformation, follows the universal wall law, remains challenging. The current paper builds upon Danis-Durbin's practice of modifying the $\omega$ equation and proposes a simple modification to the $k-\omega$ two-equation model. The formulation of the proposed modification involves dimensional analysis and the proper selection of the local length scale. The newly introduced modification is used to modify the slope of the velocity profile starting from the viscous layer to above. It recovers a semi-local scaling of turbulent kinetic energy, viscosity, and eddy frequency, then achieves a very decent correction of the velocity profile in compressible turbulent channel flows, satisfying the universal wall law after applying Trettel \& Larsson's transformation. The proposed new $k-\omega$ model can also improve the velocity and temperature predictions in strongly wall-cooled zero-pressure-gradient hypersonic turbulent boundary layers, compared to the original $k-\omega$ model. Validation using the favorable and adverse pressure gradient boundary layers suggests that the model does not impose a negative effect on the original $k-\omega$ model. Show Chinese abstract

Abstract: 在超音速和高超音速流中，由于壁面冷却引起的近壁密度变化，使用经典的涡粘性湍流模型准确预测近壁速度和温度分布是一个挑战。可压缩湍流边界层已知通过半局部变换遵循普遍壁面定律。然而，开发一种能够通过半局部变换遵循普遍壁面定律的速度分布的湍流模型仍然具有挑战性。本文基于Danis-Durbin对$\omega$方程的修改实践，并提出了对$k-\omega$双方程模型的简单修改。所提出的修改形式涉及量纲分析和局部长度尺度的正确选择。新引入的修改用于从粘性层到上方的速度分布斜率的修改。它恢复了湍流动能、粘度和涡频的半局部尺度，然后在可压缩湍流通道流中实现了非常良好的速度分布修正，在应用Trettel & Larsson变换后满足普遍壁面定律。与原始的$k-\omega$模型相比，所提出的新型$k-\omega$模型还可以改善强壁面冷却的零压梯度高超音速湍流边界层中的速度和温度预测。使用有利和不利压力梯度边界层进行验证表明，该模型不会对原始的$k-\omega$模型产生负面影响。