图像测速中直接压力梯度积分的误差传播及基于亥姆霍兹-霍奇分解（Helmholtz-Hodge Decomposition）的压力场重建方法
Error propagation of direct pressure gradient integration and a Helmholtz-Hodge decomposition based pressure field reconstruction method for image velocimetry
AI总结：
本研究分析了图像测速中直接压力梯度积分的误差传播规律，建立了PGI和PPE的误差标度律，提出基于HHD的压力场重建方法，采用带无散/无旋核的RBF实现约束校正，经合成数据验证具备高精度与灵活性。
摘要：
从图像测速测量结果中恢复压力场有两种通用策略：i）直接对动量方程得到的压力梯度进行积分；ii）求解或强制执行压力泊松方程（压力梯度的散度）。在本研究中，我们分析了前一种策略的误差传播规律，并给出了一些实用见解。例如，我们建立了压力梯度积分（PGI）和压力泊松方程（PPE）的误差标度律。我们解释了为何应用亥姆霍兹-霍奇分解（HHD）能够显著降低PGI的误差传播。我们还提出了一种新颖的基于HHD的压力场重建策略，该策略具备以下优势或特性：i）可有效处理复杂域上含噪声的离散或结构化图像测速数据；ii）采用带有无散/无旋核的径向基函数（RBFs），为不可压缩流动的速度场提供无散校正，为压力梯度提供无旋校正；iii）无需使用拉格朗日乘子即可强制执行无散/无旋约束。彻底消除测量压力梯度中的无散偏差和测量速度场中的无旋偏差后，可获得更高的精度。我们基于精确解和高保真模拟生成了合成测速数据，用以验证上述分析，同时证明RBF-HHD求解器的灵活性和有效性。
英文摘要：
Recovering pressure fields from image velocimetry measurements has two general strategies: i) directly integrating the pressure gradients from the momentum equation and ii) solving or enforcing the pressure Poisson equation (divergence of the pressure gradients). In this work, we analyze the error propagation of the former strategy and provide some practical insights. For example, we establish the error scaling laws for the Pressure Gradient Integration (PGI) and the Pressure Poisson Equation (PPE). We explain why applying the Helmholtz-Hodge Decomposition (HHD) could significantly reduce the error propagation for the PGI. We also propose to use a novel HHD-based pressure field reconstruction strategy that offers the following advantages or features: i) effective processing of noisy scattered or structured image velocimetry data on a complex domain; ii) using Radial Basis Functions (RBFs) with divergence/curl-free kernels to provide divergence-free correction to the velocity fields for incompressible flows and curl-free correction for pressure gradients; and iii) enforcing divergence/curl-free constraints without using Lagrangian multipliers. Complete elimination of divergence-free bias in measured pressure gradient and curl-free bias in the measured velocity field results in superior accuracy. Synthetic velocimetry data based on exact solutions and high-fidelity simulations are used to validate the analysis as well as demonstrate the flexibility and effectiveness of the RBF-HHD solver.
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