This works deals with sensitivity analysis for the Navier-Stokes equations. The aim is to provide an estimate of the variance of the velocity field when some of the parameters are uncertain and then to use the variance to compute confidence intervals for the output of the model. First, we introduce the physical model and analyse its stability. The sensitivity equations are derived, and their stability analysed as well. We propose a finite element-volume numerical scheme for the state and the sensitivity, which is integrated into the open-source industrial code TrioCFD. Finally, we present some numerical results: a steady and an unsteady test case for the channel flow problem are investigated. For the steady case, we compare the results to the Monte Carlo method and show how the sensitivity analysis technique succeeds in providing very accurate estimates of the variance. For the unsteady case, a new filtering procedure is proposed to deal with a sensitivity that grows in time. The filtered sensitivity is then used to compute the variance of the output and to provide confidence intervals.

中文翻译：

---

应用于不确定性传播的纳维-斯托克斯方程的灵敏度方程方法

这适用于 Navier-Stokes 方程的灵敏度分析。目的是在某些参数不确定时提供速度场方差的估计，然后使用方差计算模型输出的置信区间。首先，我们介绍物理模型并分析其稳定性。导出了灵敏度方程，并分析了它们的稳定性。我们为状态和灵敏度提出了一种有限元体积数值方案，并将其集成到开源工业代码 TrioCFD 中。最后，我们提出了一些数值结果：研究了通道流动问题的稳态和非稳态测试用例。对于稳态情况，我们将结果与蒙特卡罗方法进行比较，并展示灵敏度分析技术如何成功地提供非常准确的方差估计。对于不稳定的情况，提出了一种新的过滤程序来处理随时间增长的敏感性。然后使用过滤后的灵敏度来计算输出的方差并提供置信区间。