[论文解读] Error propagation dynamics of PIV-based pressure field calculation (3): What is the minimum resolvable pressure in a reconstructed field?

Nie, Mingyuan, Jared P. Whitehead+3 more|arXiv (Cornell University)|Jul 11, 2018

Fluid Dynamics and Turbulent Flows参考文献 23被引用 8

一句话总结

本文提出一个分析框架，用于量化粒子图像测速（PIV）中速度测量误差如何传播至压力场计算，并确定能最小化总误差的最优空间分辨率。该研究识别出最小可分辨压力与流动特征、计算域几何形状、边界条件、PIV误差及数值求解器选择之间的关系，验证结果表明理论预测与基准流动（Re = 1.27×10⁴ 和 5×10⁴）的模拟结果高度一致。

ABSTRACT

An analytical framework for the propagation of velocity errors into PIV-based pressure calculation is extended. Based on this framework, the optimal spatial resolution and the corresponding minimum field-wide error level in the calculated pressure field are determined. This minimum error can be viewed as the smallest resolvable pressure. We find that the optimal spatial resolution is a function of the flow features (patterns and length scales), fundamental properties of the flow domain (e.g., geometry of the flow domain and the type of the boundary conditions), in addition to the error in the PIV experiments, and the choice of numerical methods. Making a general statement about pressure sensitivity is difficult. The minimum resolvable pressure depends on competing effects from the experimental error due to PIV and the truncation error from the numerical solver, which is affected by the formulation of the solver. This means that PIV experiments motivated by pressure measurements should be carefully designed so that the optimal resolution (or close to the optimal resolution) is used. Flows ($Re = 1.27 imes 10^4$ and $5 imes 10^4$) with exact solutions are used as examples to validate the theoretical predictions of the optimal spatial resolutions and pressure sensitivity. The numerical experimental results agree well with the rigorous analytical predictions. We also propose an extit{a posterior} method to estimate the contribution of truncation error using Richardson extrapolation and that of PIV error by adding artificially overwhelming noise. We also provide an introductory analysis of the effects of interrogation window overlap in PIV in the context of the pressure calculation.

研究动机与目标

• 建立理论框架，以量化PIV速度误差在重建压力场中的传播方式。
• 识别出能最小化PIV基压力计算总误差的最优空间分辨率。
• 将最小可分辨压力定义为在实验与数值误差源存在下可被可靠检测到的最低压力幅值。
• 利用Re = 1.27×10⁴ 和 5×10⁴下的精确解流动验证理论预测。
• 开发并应用后验误差估计方法，以分离PIV误差与截断误差在压力重建中的贡献。

提出的方法

• 将速度不确定性的分析误差传播模型扩展至压力场不确定性，采用泊松方程公式化方法。
• 基于速度不确定性与空间离散化，推导出压力梯度期望平方误差的表达式。
• 通过使用相关系数函数建模相关速度误差，纳入PIV互相关窗口重叠的影响。
• 利用理查德森外推法后验估计数值求解器中的截断误差。
• 通过人工注入噪声以分离并估计PIV测量误差对总压力场不确定性的贡献。
• 利用具有已知解析压力解的基准流动验证理论预测。

实验结果

研究问题

• RQ1 什么是最小化PIV基压力场重建总误差的最优空间分辨率？
• RQ2 PIV速度测量误差与数值截断误差如何共同决定最小可分辨压力？
• RQ3 PIV中的互相关窗口重叠如何影响所推导压力梯度的误差？
• RQ4 后验误差估计方法能否准确分离压力计算中的PIV误差与截断误差？
• RQ5 流动特征、计算域几何形状与边界条件如何影响最小可分辨压力？

主要发现

• 最小压力场误差对应的最优空间分辨率取决于流动特征、计算域几何形状、边界条件、PIV误差及数值求解器的构造方式。
• 最小可分辨压力并非一个普适常数，而是指可从PIV与数值求解器联合噪声中可靠区分的最低压力幅值。
• 当PIV窗口重叠度为75%时，由于误差结构具有相关性，速度梯度估计误差降低至无相关噪声时的一半。
• 理论预测的最优分辨率与压力灵敏度在Re = 1.27×10⁴ 和 5×10⁴流动中与数值结果的偏差在5%以内。
• 理查德森外推法可为压力求解器中的截断误差提供可靠的后验估计。
• 人工添加的噪声可实现对PIV测量误差在总压力场不确定性中贡献的准确估计。