Abstract This paper proposes a variational formulation with a smoothness constraint for extraction of pressure from two-dimensional velocity field. The method of characteristics is first described as a reduced case to obtain pressure data by a line integral along a given line. Then, a global optimization method is developed, and the Euler–Lagrange equation is obtained to determine a pressure field from velocity data in a domain with suitable boundary conditions. Numerical algorithms are developed to solve the Euler–Lagrange equation with suitable boundary conditions. The accuracy of the developed methods and the effects of the relevant parameters on extraction of pressure are evaluated on a simulated two-dimensional oblique Hiemenz flow. Subsequently, in an experimental case, pressure fields near a freely flying hawkmoth are estimated from high-resolution velocity fields that are obtained by applying the optical flow method to high-speed Schlieren photography images. Graphic abstract Pressure field near the hawkmoth wingtip obtained by using (a) Direct-QR method. And (b) iterative method.

中文翻译：

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从二维速度确定压力的变分法

摘要 本文提出了一种带有平滑约束的变分公式，用于从二维速度场中提取压力。特征方法首先被描述为通过沿给定线的线积分获得压力数据的简化情况。然后，开发了一种全局优化方法，并获得了欧拉-拉格朗日方程，以根据具有合适边界条件的域中的速度数据确定压力场。开发了数值算法来求解具有合适边界条件的 Euler-Lagrange 方程。所开发方法的准确性和相关参数对压力提取的影响在模拟的二维倾斜 Himenz 流上进行评估。随后，在一个实验案例中，自由飞行的天蛾附近的压力场是根据高分辨率速度场估计的，该速度场是通过将光流方法应用于高速纹影摄影图像而获得的。使用 (a) Direct-QR 方法获得的天蛾翼尖附近的图形抽象压力场。(b) 迭代法。