A regularization method for delivering the fourth-order derivative of experimental data and its applications in fluid-structure interactions

In the experimental investigation of fluid-structure interactions regarding the undulatory motion like flag flapping or fish swimming, solving the force distribution on the flexible body stands as an indispensable endeavor to gain insights into the underlying dynamic mechanisms. However, the solving process entails high-order numerical derivatives of experimental data, which poses a formidable challenge for experimental studies on fluid-structure interactions, given that the measurement noise inherent in experimental data renders the problem ill-posed. The commonly practiced regularization methods for numerical derivatives are feeble to tackle the fourth-order derivative associated with the bending force; those methods, in particular, require predetermined parameters about the unknown noise. We introduce here an empirical regularization method founded upon the kernel-term modification in the frequency domain, notably capable of determining the fourth derivative of experimental data. By leveraging the potentials of the iterative operations, our method enables the reliable estimation of an approximately optimal regularization parameter, all without reliance on any a priori knowledge about the noise characteristics. To demonstrate the reliability, robustness, and accuracy of the method, we perform rigorous numerical assessments using different data models that are infused with noise varying several orders of magnitude. Additionally, practical application of this method is achieved in the experiment on a flexible film flapping in the gusty flow, where the spatiotemporal distribution of the bending force density on the film is calculated by integrating this method with a linear reconstruction.