Inverse Reconstruction of Free-Stream Vortices Using Surface Pressure on a Joukovsky Airfoil

We investigate the potential flow around a Joukowsky airfoil and demonstrate how sparse surface pressure measurements can be used to infer upstream flow features through a fully analytical framework. Leveraging the conformal mapping between a circular cylinder and the airfoil shape, we utilize the closed-form expressions for the velocity and pressure distribution across the surface. This mapping enables us to directly perform sensitivity analysis of surface pressure readings to the external vortex fields. By analytically computing the sensitivity field from individual pressure sensors, named as their domains of dependence, we identify the likely source and trajectory of upstream disturbances that influence the flow near the airfoil. The posterior distribution of the location of the freestream vortex is analytically derived, and numerically computed and visualized, given a fixed level of measurement noise. We find that the proper sensor layout is crucial to reduce the uncertainty in the located vortex. The results highlight the power of classical complex analysis in providing insight into flow reconstruction and sensing strategies with uncertainty quantification, offering a foundational approach for educational, diagnostic, and real-time aerodynamic applications.