Uncertainty in the critical amplification factor ( $N_{cr}$ ) of the $e^N$ transition model is used to approximate the uncertainty in the surface and flow quality of natural laminar flow (NLF) aerofoils. The uncertainty in $N_{cr}$ is represented by a negative half-normal probability distribution that descends from the largest $N_{cr}$ achievable with an ideal surface and flow quality. The uncertainty in various aerodynamic coefficients due to the uncertainty in $N_{cr}$ is quantified using the weighted mean and standard deviation of flow solutions run at different $N_{cr}$ values. The uncertainty in the aerofoil performance is assessed using this methodology. It is found that the standard deviation of the aerofoil performance due to the uncertainty in $N_{cr}$ is largest when the transition location is most sensitive to changes in the lift coefficient at the ideal $N_{cr}$ . Robust shape optimisation is also carried out to improve the mean performance and reduce the standard deviation of the performance with uncertainty in $N_{cr}$ . This is found to be effective at producing aerofoils with a larger amount of laminar flow that are less sensitivity to uncertainty in $N_{cr}$ . A trade-off is observed between the mean performance and the standard deviation of the performance. It is also found that reducing the standard deviation of the performance at one Mach number or lift coefficient design point can cause an increase in the standard deviation off-design.

中文翻译：

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层流翼型的不确定性分析和稳健形状优化

关键放大因子的不确定性 ($N_{cr}$） 的$e^N$过渡模型用于近似自然层流（NLF）机翼的表面和流动质量的不确定性。中的不确定性$N_{cr}$由一个负半正态概率分布表示，该分布从最大的$N_{cr}$可实现理想的表面和流动质量。各种空气动力系数的不确定性，由于不确定性$N_{cr}$使用不同运行的流动解决方案的加权平均值和标准偏差进行量化$N_{cr}$价值观。使用这种方法评估翼型性能的不确定性。发现由于不确定性导致翼型性能的标准偏差$N_{cr}$当过渡位置对理想升力系数的变化最敏感时最大$N_{cr}$. 还进行了稳健的形状优化以提高平均性能并减少具有不确定性的性能标准偏差$N_{cr}$. 发现这对于生产具有大量层流的翼型是有效的，这些翼型对不确定性的敏感性较低$N_{cr}$. 在平均性能和性能的标准偏差之间观察到权衡。还发现，降低一个马赫数或升力系数设计点的性能标准偏差会导致偏离设计的标准偏差增加。