Improved Approximations to Wagner Function Using Sparse Identification of Nonlinear Dynamics

Abstract

The Wagner function represents the response in lift on an airfoil that is subject to a sudden change in conditions. While it plays a fundamental role in the development and application of unsteady aerodynamic methods, explicit expressions for this function are difficult to obtain. This has led to numerous proposed approximations that facilitate convenient and rapid computations. While these approximations can be sufficient for many purposes, their behavior is often noticeably different from the true Wagner function, especially for long-time asymptotic behavior. In particular, while many approximations have small maximum absolute error across all times, the relative error of the asymptotic behavior can be substantial. Here, we propose an alternative approximation methodology that is accurate for all times, for a variety of accuracy measures. This methodology casts the Wagner function as the solution of a nonlinear scalar ordinary differential equation, which is identified using a variant of the sparse identification of nonlinear dynamics algorithm. We show that this approach yields accurate approximations using either first- or second-order differential equations. We additionally demonstrate that this method can be applied to model the lift response for a more realistic aerodynamic system, featuring a finite thickness airfoil and nonplanar wake.