An inverse method for reconstruction of the in situ steady-state hydrodynamic load distribution using limited strain measurements is developed and validated on a rectangular, flexible surface-piercing hydrofoil in fully-wetted (FW) and fully-ventilated (FV) flows. The hydrofoil is used as a canonical proxy to more complex hydrodynamic lifting surfaces such as marine propulsors and turbines. The approach involves using a forward fluid–structure interaction (FSI) model to predict the hydroelastic response for given operating conditions. The inverse problem is solved as an optimization problem to determine unknown operating conditions. The forward FSI model consists of a nonlinear lifting line (LL) fluid solver with considerations for free surface, ventilation, and viscous effects, and a solid finite element method (FEM) solver using 1-D beam elements representing the spanwise bending and twisting deformations.
The coupled FSI model was validated using data collected during towing tank experiments at the University of Michigan. Predictions of the lift and moment coefficients, as well as spanwise bending and twisting deformations agreed well with experimental results in FW and in FV flows.
The inverse problem is formulated as an optimization problem to determine the unknown operating conditions that will minimize the difference between the measured and predicted deformations. To avoid non-uniqueness problems often encountered by inverse problems, a dynamic constraint using the measured wetted natural frequencies was added to help regularize the problem and speed up the solution process. A sequential quadratic programming algorithm was used as the optimizer for the inverse problem. The experimental studies showed that the inverse FSI model accurately determined the unknown operating conditions (angle of attack and immersed aspect ratio) for a given a known flow speed and a limited number of strain measurements in both FW and FV conditions. The converged results were used to reconstruct a 3-D hydrodynamic load distribution on the foil and to predict the cavity shape for FV operating conditions, which were found to agree well with experimental measurements and observations.